Cengage Maths Solutions Class 12 Application of Derivatives - Calculus

CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER APPLICATION CALCULUS

OF

DERIVATIVES

||

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Question CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals Find the total number of parallel tangents of f 1 (x) = x 2 - x + 1andf 2 (x) = x 3 - x 2 - 2x + 1.

1

f1 (x) = x2 − x + 1andf2 (x) = x3 − x2 − 2x + 1.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 2

Prove that the tangent drawn at any point to the curve f(x) = x5 + 3x3 + 4x + 8 would make an acute angle with the x-axis.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 3

| |

Find the equation of the normal to the curve y = x 2 - x ∣ ∣ atx = - 2.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 4

Find the equation of tangent to the curve y =

sin - 1(2x) 1+x

2

atx =

√3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

Find the equation of the tangent to the curvey =

{

x2

sin1 x

, x ≠ 00, x = 0ahe or ig ∈

5

sin 1 curvey = {x , x ≠ 00, x = 0ahe or ig x ∈ 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals Find the equation of tangent and normal to the curve x =

2at 2

(1 + t ) 2

6 point for which t =

1 2

,y =

2at 3

(1 + t )

at the

2

.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 7

Find the equations of the normal to the curve y = x 3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 8

For the curve y = 4x 3 - 2x 5, find all the points at which the tangents pass through the origin.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

9

(

)

If the tangent at any point 4m 2 , 8m 2 of x 3 - y 2 = 0 is a normal to the curve x 3 - y 2 = 0 , then find the value of m.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 10

For the curve xy = c, prove that the portion of the tangent intercepted between the coordinate axes is bisected at the point of contact.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 11

Find the equation of all possible normals to the parabola x 2 = 4y drawn from the point (1, 2).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 12

Find the point on the curve where tangents to the curve y 2 - 2x 3 - 4y + 8 = 0 pass through (1,2).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 13

Find the equation of the normal to the curve x 3 + y 3 = 8xy at the point where it meets the curve y 2 = 4x other than the origin.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 14

Show that the straight line xcosα + ysinα = p touches the curve xy = a 2 , if p 2 = 4a 2cosαsinα.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

(

If the line xcosθ + ysinθ = P is the normal to the curve (x + a)y = 1, then show θ ∈ 2nπ + 15

θ ∈ (2nπ +

π , (2n + 1)π) 2 3π ∪ (2nπ + , (2n + 2)π), n ∈ Z 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals Show that the tangent to the curve 3xy 2 - 2x 2 y = 1at(1, 1) meets the curve again at the 16

(

point -

16 5

, -

1 20

)

.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 17

Find the normal to the curve x = a(1 + cosθ), y = asinθahη. Prove that it always passes through a fixed point and find that fixed point.  Watch Free Video Solution on Doubtnut

π 2

,

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

18

If the curve y = ax 2 - 6x + b pass through (0, 2) and has its tangent parallel to the x-axis 3 at x = , then find the values of aandb.

2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 19

(

)

Find the equation of the tangent to the curve 1 + x 2 y = 2 - x, where it crosses the xaxis.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 20

If the equation of the tangent to the curve y 2 = ax 3 + b at point (2, 3)isy = 4x - 5 , then find the values of aandb .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

21

Find the value of n ∈ N such that the curve x a

+

y b

() () x n a

+

y n b

= 2 touches the straight line

= 2 at the point (a, b).

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 22

Find the condition that the line Ax + By = 1 may be normal to the curve a n - 1y = x n.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 23

If the tangent to the curve xy + ax + by = 0 at (1, 1) is inclined at an angle tan - 1 2 with xaxis, then find aandb ?  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 24

If the tangent at (1, 1) on y 2 = x(2 - x) 2 meets the curve again at P, then find coordinates of P.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

25

(

)

Does there exists line/lines which is/are tangent to the curve y = sinxat x 1 , y 1 and

(

)

normal to the curve at x 2 , y 2 ?  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 26

In the curve x ay b = K a + b , prove that the potion of the tangent intercepted between the coordinate axes is divided at its points of contact into segments which are in a constant ratio. (All the constants being positive).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal 27

Find the length of sub-tangent to the curve y = ex / a  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal 28

Determine p such that the length of the such-tangent and sub-normal is equal for the curve y = e px + px at the point (0, 1).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal 29

Find the length of normal to the curve x = a(θ + sinθ), y = a(1 - cosθ) at θ =

π 2

.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal 30

In the curve x m + n = am - ny 2n , prove that the mth power of the sub-tangent varies as the nth power of the sub-normal.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal 31

Find the length of the tangent for the curve y = x 3 + 3x 2 + 4x - 1 at point x = 0.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal 32

(

)

For the curve y = a1n x2 - a 2 , show that the sum of length of tangent and subtangent at any point is proportional to product of coordinates of point of tangency.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal 33

For the curve y = f(x) prove that (lenght n or mal)^2/(lenght or tanght)^2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal 34

If the sub-normal at any point on y = a 1 - nx n is of constant length, then find the value of n.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 35

OF

DERIVATIVES_Angle

Find the cosine of the angle of intersection of curves f(x) = 2 x(log) exandg(x) = x 2x - 1.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 36

x2

OF

DERIVATIVES_Angle

y2

Find the value of a if the curves 2 + = 1andy 3 = 16x cut orthogonally.

4 a  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 37

|

OF

|

DERIVATIVES_Angle

|

|

Find the acute angle between the curves y = x2 - 1 and y = x 2 - 3 at their points of

intersection.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 38

OF

DERIVATIVES_Angle

Find the angle at which the two curves x 3 - 3xy 2 + 2 = 0and3x 2 y - y 3 - 2 = 0 intersect.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 39

OF

DERIVATIVES_Angle

OF

DERIVATIVES_Angle

Find the angle of intersection of y = a xandy = bx  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 40

Find the angle of intersection of the curves xy = a 2 andx 2 + y 2 = 2a 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 41

OF

DERIVATIVES_Angle

Find the angle at which the curve y = Ke Kx intersects the y-axis.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 42

OF

DERIVATIVES_Angle

If the curves ay + x 2 = 7andx 3 = y cut orthogonally at (1, 1) , then find the value a.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 43

OF

DERIVATIVES_Angle

2 y Find the angle between the curves x 2 = a 2andC 2 : xy 3 = c 3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 44

OF

DERIVATIVES_Angle

Find the angle between the curves 2y 2 = x 3andy 2 = 32x.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 45

OF

Find the possible values of p such that the equation px 2 = (log) ex has exactly one solution.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications

46

Find the values of a if equation 1 - cosx = solution.

√3 2

OF

|x| + a, x ∈ (0, π), has exactly one

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications

47

(

Find the locus of point on the curve y 2 = 4a x + as ∈ the axis of x.  Watch Free Video Solution on Doubtnut

OF

x a

)

where tangents are parallel to

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 48

OF

Find the shortest distance between the line y = x - 2 and the parabola y = x 2 + 3x + 2.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 49

OF

Find the point on the curve 3x 2 - 4y 2 = 72 which is nearest to the line 3x + 2y + 1 = 0.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 50

OF

The tangent at any point on the curve x = acos 3 θ, y = asin3 θ meets the axes in PandQ .

Prove that the locus of the midpoint of PQ is a circle.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 51

Prove that all the point on the curve y = axis lie on parabola.

OF

√x + sinx at which the tangent is parallel to x-

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 52

OF

Find the distance of the point on y = x 4 + 3x2 + 2x which is nearest to the line y = 2x - 1  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION

OF

DERIVATIVES_Miscellaneous Applications 53

The graph y = 2x 3 - 4x + 2andy = x 3 + 2x - 1 intersect in exactly 3 distinct points. Then find the slope of the line passing through two of these points.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications

54

Find the minimum value of

(x 1 - x 2 ) 2 +

(

x 21 20

OF

-

√ (17 - x 2 )(x 2 - 13 )

)

2

where

x 1 ∈ R + , x 2 ∈ (13, 17).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure 55

1 3 t - 6t . Find the acceleration 2 at the time when the velocity vanishes (i.e., velocity tends to zero).

Displacement s of a particle at time t is expressed as s =

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure 56

On the curve x 3 = 12y, find the interval of values of x for which the abscissa changes at a faster rate than the ordinate?  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure m

57

The length x of a rectangle is decreasing at the rate of 5c and the width y is m m increasing at the rate of 4c When x = 8cm and y = 6cm, find the rate of change of (a) m the perimeter and (b) the area of the rectangle.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure 58

The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3cm/s . How fast is the area decreasing when the two equal sides are equal to the base?  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure

59

A lamp is 50ft. above the ground. A ball is dropped from the same height from a point 30ft . away from the light pole. If ball falls a distance s = 16t 2 ft . in t second, then how 1 fast is the shadow of the ball moving along the ground s later?

2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure 60

If water is poured into an inverted hollow cone whose semi-vertical angel is 30 0, show that its depth (measured along the axis) increases at the rate of 1 cm/s. Find the rate at which the volume of water increases when the depth is 24 cm.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure

61

Let x be the length of one of the equal sides of an isosceles triangle, and let θ be the angle between them. If x is increasing at the rate (1/12) m/h, and θ is increasing at the π rate of radius/h, then find the rate in m 3 / h at which the area of the triangle is 180 increasing when x = 12mandthη = π /4.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure

62

A horse runs along a circle with a speed of 20km/h . A lantern is at the centre of the circle. A fence is along the tangent to the circle at the point at which the horse starts.

Find the speed with which the shadow of the horse moves along the fence at the moment when it covers 1/8 of the circle in km/h.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure 63

The distance covered by a particle moving in a straight line from a fixed point on the line is s, where s 2 = at 2 + 2bt + ⋅ Then prove that acceleration is proportional to s - 3 .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure 64

Tangent of an angle increases four times as the angle itself. At what rate the sine of the angle increases w.r.t. the angle?  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure 65

Two cyclists start from the junction of two perpendicular roads, there velocities being 3um/ m ∈ and 4um /m ∈ , respectively. Find the rate at which the two cyclists separate.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure 66

A spherical iron ball 10cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50cm 3 / m ∈ . When the thickness of ice is 5cm, then find the rate at which the thickness of ice decreases.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure 67

xandy are the sides of two squares such that y = x - x 2 . Find the rate of the change of the area of the second square with respect to the first square.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure 68

Two men PandQ start with velocity u at the same time from the junction of two roads inclined at 45 0 to each other. If they travel by different roads, find the rate at which they are being separated.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Approximations 69

OF

1

Find the approximate value of(0. 0037) 2 .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Approximations 70

OF

1

Find the approximate value of (26) 3 .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Approximations 71

OF

Find the approximate change in the volume V of a cube of side x meters caused by increasing side by 1 % .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Approximations 72

OF

Find the approximate value of f(5. 001), where f(x) = x3 - 7x 2 + 15.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Approximations

OF

In an acute triangle ABC if sides a, b are constants and the base angles AandB vary, 73

then show that

dA

√

a2

-

b 2sin 2A

=

dB b 2 - a2 sin2 B

√

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Approximations 74

OF

Find the approximate value of f(3. 02), where f(x) = 3x2 + 5x + 3.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Approximations 75

OF

If the radius of a sphere is measured as 9cm with an error of 0.03 cm, then find the approximate error in calculating its volume.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Approximations 76

OF

Find the approximate value of (1. 999) 6 .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Approximations 77

OF

If 1 0 = α radians, then find the approximate value of cos60 01 ′ .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Approximations

OF

78

If in a triangle ABC, the side c and the angle C remain constant, while the remaining da db elements are changed slightly, show that + = 0. cosA cosB  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Approximations

79

OF

Discuss the applicability of Rolles theorem for the following functions on the indicated intervals: f(x) = |x| ∈ [ - 1, 1] f(x) = 3 + (x - 2) 2 / 3 in [1,3] f(x) = tanξn[0, π] f(x) = log

{ } x 2 + ab

x(a + b)

in `[a , b],w h e r e-

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems →

80

If f : [ - 5, 5] R is differentiable function and iff ′ (x) does not vanish anywhere, then prove that f( - 5) ≠ f(5) .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

81

If the function f(x) = x 3 - 6x 2 + ax + b defined on [1,3] satisfies Rolles theorem for 2√3 + 1 c= then find the value of aandb √3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 82

How many roots of the equation (x - 1)(x - 2)(x - 3) + (x - 1)(x - 2)(x - 4) + (x - 2)(x - 3)(x - 4) + (x - 1)(x - 3)(x - 4) = 0 are positive?  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 83

Let f(x) be differentiable function and g(x) be twice differentiable function. Zeros of f(x), g ′ (x) be a, b , respectively, `(a  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

84

If φ(x) is differentiable function ∀ x ∈ R and a ∈ R + such that φ(0) = φ(2a), φ(a) = φ(3a)andφ(0) ≠ φ(a) then show that there is at least one root of equation φ ′ (x + a) = φ ′ (x) ∈ (0, 2a)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 85

If 2a + 3b + 6c = 0, then prove that at least one root of the equation ax2 + bx + c = 0 lies in the interval (0,1).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 86

Show that between any two roots of e - x - cosx = 0, there exists at least one root of sinx - e - x = 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 87

Let P(x) be a polynomial with real coefficients, Let `a , b in R ,a  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 88

If the equation ax 2 + bx + c = 0 has two positive and real roots, then prove that the equation ax 2 + (b + 6a)x + (c + 3b) = 0 has at least one positive real root.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

89

Let f(x)andg(x) be differentiable functions such that f ′ (x)g(x) ≠ f(x)g ′ (x) for any real x.

Show that between any two real solution of f(x) = 0, there is at least one real solution of g(x) = 0.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 90

Consider the function f(x) = 8x 2 - 7x + 5 on the interval [ - 6, 6] . Find the value of c that satisfies the conclusion of Lagranges mean value theorem.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 91

Let f be differentiable for all x, If f(1) = - 2andf ′ (x) ≥ 2 for all x ∈ [1, 6], then find the range of values of f(6) .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

92

→ Let f: [2, 7]0, ∞ be a continuous and differentiable function. Then show that (f(7)) 2 + (f(2)) 2 + f(2)f(7) (f(7) - f(2)) = 5f 2 (c)f ′ (c), where c ∈ [2, 7].

3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value

Theorems 93

Using Lagranges mean value theorem, prove that |cosa - cosb| ≤ |a - b|.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

94

Let f(x)andg(x) be differentiable function in (a, b), continuous at aandb, andg(x) ≠ 0 in g(a)f(b) - f(a)g(b) (b - a)g(a)g(b) [a, b]. Then prove that = g(c)f ′ (c) - f(c)g ′ (c) (g(c)) 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

95

Using β -α

mean

value

(

theorem,

show

that

)

- 1 β - tan - 1 α < (β - α) 1 + α 2 , β > α > 0. < tan 1 + β2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 96

Let f(x)andg(x) be two differentiable functions in Randf(2) = 8, g(2) = 0, f(4) = 10, andg(4) = 8. Then prove that g ′ (x) = 4f ′ (x) for at least one x ∈ (2, 4).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 97

Suppose α, βandthη are angles satisfying `0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

98

Let f be continuous on [a, b], a > 0, and differentiable on (a, b) . Prove that there exists bf(a) - af(b) c ∈ (a, b) such that = f(c) - cf ′ (c) b-a  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 99

Find the condition if the equation 3x 2 + 4ax + b = 0 has at least one root in (0, 1) .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 100

Find c of Lagranges mean value theorem for the function f(x) = 3x 2 + 5x + 7 in the interval [1, 3].  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 101

Let `0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 102

Let f(x)andg(x) be differentiable for 0 ≤ x ≤ 2 such that f(0) = 2, g(0) = 1, andf(2) = 8. Let there exist a real number c in [0, 2] such that f ′ (c) = 3g ′ (c). Then find the value of g(2) .

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 103

Prove that if 2a02 < 15a, all roots of x5 - a 0x 4 + 3ax 3 + bx 2 + cx + d = 0 cannot be real. It is given that a 0, a, b, c, d ∈ R.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

104

If f(x) is continuous in [a, b] and differentiable in (a,b), then prove that there exists at f ′ (c) f(b) - f(a) least one c ∈ (a, b) such that 2 = 3c b3 - a3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 105

|

|

Prove that tan - 1 x - tan - 1 y ≤ |x - y| ∀ x, y ∈ R.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

106

Using Lagranges mean value theorem, prove that

b -a b

< log

() b a

<

b-a a

= a, where

0 < a < b.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

107

If a > b > 0, with the aid of Lagranges mean value theorem, prove that nb n - 1 (a - b) < a n - b n < na n - 1(a - b), if n > 1. nb n - 1 (a - b) > a n - b n > na n - 1(a - b), if 0 < n < 1.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 108

Let f(x)andg(x) be two functions which are defined and differentiable for all x ≥ x 0. If

( ) ( )

f x0 = g x 0 andf ′ (x) > g ′ (x) for all x > x0 , then prove that f(x) > g(x) for all x > x 0 .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 109

Iff(x)andg(x) be two function which are defined and differentiable for all x ≥ x 0. If

( ) ( )

f x0 = g x 0 andf ′ (x) > g ′ (x) for all f > x 0, then prove that f(x) > g(x) for all x > x 0.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 110

If f(x)andg(x) are continuous functions in [a, b] and are differentiable in(a, b) then prove that there exists at least one c ∈ (a, b) for which. `|f(a)f(b)g(a)g(b)|=(b-a)|f(a)f^(prime) (c)g(a)g^(prime)(c)|,w h e r ea  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions Find 111

the

complete

set

of

values

of

λ,

OF for

which

the

function

f(x) =

{x + 1, x < 1λ, x = 1x

2-

x + 3, x > 1 is strictly increasing at x = 1

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 112

OF

Prove that the function are increasing for the given intervals: f(x) = e x + sinx, x ∈ R +  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions Prove 113

that

the

function

are

increasing

OF for

the

given

intervals:

( )

f(x) = sinx + tanx - 2x, x ∈ 0,

π 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions Prove 114

that

the

function

are

increasing

OF for

the

given

intervals:

( )

f(x) = secx - cosecx, x ∈ 0,

π 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 115

OF

Find the least value of k for which the function x 2 + kx + 1 is an increasing function in the interval 1 < x < 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

[ [

→

116

If f : 0, ∞ R is the function defined by f(x) = is injective or not.

OF

ex ^ 2 - e - x ^ 2 x

e ^ 2+e

-x

^2

, then check whether f(x)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

OF →

117

Let f(x)andg(x) be two continuous functions defined from RR , such that `f(x_1)>f(x_2)a n dg(x_1)f(g(3alpha-4))`  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 118

(

OF

)

Prove that the function f(x) = (log) e x 2 + 1 - e - x + 1 is strictly increasing ∀ x ∈ R.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 119

OF

Prove that f(x) = x - sinx is an increasing function.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 120

Find the values of p if f(x) = cosx - 2px is invertible.

OF

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 121

OF

Find the values of a if f(x) = 2e x - ae - x + (2a + 1)x - 3 is increasing for all values of x.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 122

OF

Is every invertible function monotonic?  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

123

OF

If fogoh(x) is an increasing function, then which of the following is not possible? (a) f(x), g(x), andh(x) are increasing (b)f(x)andg(x) are decreasing and h(x) is increasing (c) f(x), g(x), andh(x) are decreasing  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

124

OF

→ → Let f: [0, ∞) 0, ∞andg: [0, ∞)0, ∞ be non-increasing and non-decreasing functions, respectively, and h(x) = g(f(x)) . If fandg are differentiable functions, h(x) = g(f(x)). If fandg are differentiable for all points in their respective domains and h(0) = 0, then show h(x) is always, identically zero.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

125

(

OF

If f x0andg(x) = f(x) 1 - 2(f(x)) 2 are strictly increasing ∀ x ∈ R, then find the values of

√

f(x).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 126

OF

f(x) = [x] is step-up function. Is it a monotonically increasing function for x ∈ R ?  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 127

OF

Separate the intervals of monotonocity of the function: f(x) = 3x 4 - 8x 3 - 6x 2 + 24x + 7  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions Separate 128

f(x) = -

sin3 x

the +

3sin 2x

intervals + 5, x ∈

of

OF

monotonocity

of

the

function:

[ ]

π π .

- , 2 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

129

OF

Find the intervals in which the function f given by f(x) =

4sinx - 2x - xcosx 2 + cosx

decreasing, x ∈ (0, 2π)

is increasing

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 130

Find the interval of monotonocity of the function f(x) =

OF |x - 1| x

2

.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 131

OF

Find the intervals of decrease and increase for the function f(x) = cos

() π

x

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 132

OF

Let g(x) = (f(x)) 3 - 3(f(x)) 2 + 4f(x) + 5x + 3sinx + 4cosx ∀x ∈ R. Then prove that g is increasing whenever is increasing.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 133

OF

Let g(x) = f(x) + f(1 - x)andf x > 0 ∀ x ∈ (0, 1) . Find the intervals of increase and decrease of g(x) .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 134

Find the number of solutions of the equation 3tanx +

OF

x3

( )

= 2 ∈ 0,

π 4

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

OF

135

Find the number of roots of the function f(x) =

1 (x + 1)

3

- 3x + sinx .

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 136

OF

Prove that the following functions are strictly increasing: f(x) = cot - 1 x + x  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 137

OF

Separate the intervals of monotonocity for the function f(x) = - 2x 3 - 9x 2 - 12x + 1  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 138

Separate the intervals of monotonocity for the function f(x) = x 2 e - x  Watch Free Video Solution on Doubtnut

OF

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 139

OF

Separate the intervals of monotonocity for the function f(x) = sinx + cosx, x ∈ (0, 2π)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 140

OF

Separate the intervals of monotonocity f(x) = 3cos 4x + 10cos 3 x + 6cos 2 x - 3, x ∈ [0, π]

for

the

function

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 141

OF

(

) (

Separate the intervals of monotonocity for the function f(x) = (log) ex 2 + (log) ex

)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 142

Discuss monotonocity of f(x) =

OF

x

and `g(x)=x/(tanx),w h e r e sinx  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

143

Discuss monotonocity of y = f(x) which is griven by x =

 Watch Free Video Solution on Doubtnut

OF 1 1 + t2

andy =

1

(

t 1 + t2

)

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 144

OF

Find the value of a for which the function (a + 2)x 3 - 3ax 2 + 9ax - 1 decreases montonically for all real x.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 145

Find the value of a in order that f(x) =

OF

√3sinx - cosx - 2ax + b decreases for all real

values of x .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 146

OF

Discuss the monotonocity of function f(x) = 2log|x - 1| - x 2 + 2x + 3.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 147

OF

Let g(x) = f(logx) + f(2 - logx)andf x < 0 ∀ x ∈ (0, 3) . Then find the interval in which g(x) increases.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 148

Find the points of inflection for f(x) = sinx f(x) = 3x4 - 4x 3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 149

Prove that ln(1 + x) < x for x < 0.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 150

Show that `0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 151

If `a , b >0a n d0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 152

Prove that |cosα - cosβ| ≤ |α - β|  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 153

For 0 < x <

π 2

, prove that cos(sinx) > sin(cosx)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 154

Let fandg be differentiable on R and suppose f(0) = g(0)andf ′ (x) ≤ g ′ (x) for all x ≥ 0.

Then show that f(x) ≤ g(x) for all x ≥ 0.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection

155

Prove that e x +

√1 + e 2x ≥ (1 + x) + √2 + 2x + x 2 ∀x ∈ R

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 156

Show that

x (1 + x)

< 1n(1 + x) for x > 0

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 157

For `0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 158

x

Show that tan - 1 x >

x2

if x ∈ (0, ∞) .

1+ 3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 159

Prove that f(x) =

sinx

[ ]

is monotonically decreasing in 0,

x  Watch Free Video Solution on Doubtnut

π 2

. Hence, prove that `(2x)/pi

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection

160 For `0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 161

(

The function f(x) = x 2 - 4

) (x n

2

)

- x + 1 , n ∈ N, assumes a local minimum value at

x = 2. Then find the possible values of n  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 162

Find the points at which the function f given by f(x) = (x - 2) 4(x + 1) 3 has local maxima local minima point of inflexion  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

163

Lert f: [a, b] → be a function such that for c ∈ (a, b), f ′ (c) = f c = f c = f iv(c) = f v(c) = 0.

Then f (a)has a local extermum at x = ⋅ f (b)has neither local maximum nor minimum at x = c f (c)is necessarily a constant function (d)it is difficult to say whether a

)

or (b).

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 164

Discuss the extremum of f(x) =

40 3x 4 + 8x3 - 18x 2 + 60

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 165

( )

Discuss the extremum of f(x) = sinx(1 + cosx), x ∈ 0,

π 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 166

Discuss the maxima/minima of the function f(x) =

4sinx - 2x - xcosx 2 + cosx

, 0x < 2π

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 167

1 Discuss the extremum of f(x) = x 2 + 2 x  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 168

Find the maximum value of f(x) =

()

1 x x

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 169

Let f(x) =

a x

+ x 2. If it has a maximum at x = - 3, then find the value of a.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 170

Discuss the extrema of f(x) =

x 1 + xtanx

( )

, x ∈ 0,

π 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 171

Discuss the extremum of f(x) = asecx + bcosecx, 0 < a < b  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

172

Find the range of the function f(x) =

x4 + x2 + 5

(

x2 + 1

)

2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

173

If the function y = f(x) is represented as x = ϕ(t) = t 5 - 5t 3 - 20t + 7, y = ψ(t) = 4t 3 - 3t 2 - 18t + 3(|t| < 2), then find the maximum and minimum values of y = f(x).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 174

If f(x) =

{x , x ≤ 02sinx, x > 0′ ∈ vestigatethe function at x = 0 for maxima/minima 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 175

f(x) =

{

cos(πx) 2

, x > 0x + a, x ≤ 0 Find the values of a if x = 0 is a point of maxima.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 176

f(x) = |ax - b| + c|x| ∀ x ∈ ( - ∞, ∞), where a > 0, b > 0, c > 0. Find the condition if f(x) attains the minimum value only at one point.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 177

2

Discuss the extremum of f(x) = 2x + 3x 3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 178

Let f(x) = 2x 3 = 9x2 + 12x + 6. Discuss the global maxima and minima of f(x) ∈ [0, 2]and(1, 3) and, hence, find the range of f(x) for corresponding intervals.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 179

Find the maximum value and the minimum value and the minimum value of 3x 4 - 8x 3 + 12x 2 - 48x + 25 on the interval [0, 3] .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum

180

Discuss the global maxima and global minima of f(x) = tan - 1(log) ex in

[ ] 1

√3

, √3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 181

Find the range of the function f(x) = 2√ x - 2 + √ 4 - x  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 182

Discuss the extremum of f(x) = 2x 3 - 3x 2 - 12x + 5 for x ∈ [ - 2, 4] and the find the range of f(x) for the given interval.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 183

Discuss the extremum of f(x) = 1 + 2sinx + 3 cos 2x, x ≤ x ≤

2π 3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 184

Discuss the extremum of f(x) = sinx +

1

sin2x +

2

1 3

sin3x, 0 ≤ x ≤ π.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum Find 185

the

maximum

and

(

minimum

)

y = (log) e 3x 4 - 2x 3 - 6x 2 + 6x + 1 ∀x ∈ (0, 2)

(3x

4

values

of Given

the

function that

)

- 2x 3 - 6x 2 + 6x 2 + 6x + 1 > 0Ax ∈ (0, 2)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum

186

Let f(x) =

- sin 3x

+

3sin2 x

[ ]

+ 5 on 0,

π 2

. Find the local maximum and local minimum

of f(x).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 187

Discuss the extremum of f(x) =

1 3

( ) x+

1 x

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 188

(

Discuss the extremum of f(x) = x x2 - 4

)

1

-3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum

189

Let f(x) =

{x

3

(

)

- x 2 + 10x - 5, x ≤ 1 - 2x + (log) 2 b2 - 2 , x > 1 Find the values of b for

which f(x) has the greatest value at x = 1.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 190

Let f(x) be defined as f(x) =

{tan

- 1α

- 5x2 , 0 < x < 1 and - 6x, x ≥ 1 if f(x) has a

maximum at x = 1, then find the values of α .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 191

Discuss the extremum of `f(x)={|x^2-2|,-1lt=x  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum

| |

192

Find the minimum value of |x| + x +

1 2

| |

+ |x - 3| + x -

5 2

.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 193

Discuss the extremum of f(x) =

{1 + sinx, x < 0x

2

- x + 1, x ≥ 0atx = 0

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 2

194

4

Discuss the maxima and minima of the function f(x) = x 3 - x 3 . Draw the graph of y = f(x) and find the range of f(x) .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum

195

The curve f(x) =

x 2 + ax + 6

has a stationary point at (4, 1) . Find the values of aandb . x - 10 Also, show that f(x) has point of maxima at this point.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials 196

Find the value of a if x 3 - 3x + a = 0 has three distinct real roots.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials 197

Prove that there exist exactly two non-similar isosceles triangles ABC such that tanA + tanB + tanC = 100.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials

198

If t is a real number satisfying the equation 2t 3 - 9t 2 + 30 - a = 0, then find the values of 1 the parameter a for which the equation x + = t gives six real and distinct values of x . x  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 199

x2 - x + 1

Draw the graph of f(x) = 2 . x +x+1  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 200

Draw the graph of f(x) =

x 2 - 5x + 6 x2 - x

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

201

Draw the graph of f(x) =

| | x2 - 2 2

x -1

.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum Two towns AandB are 60 km apart. A school is to be built to serve 150 students in

202

town Aand50 students in town B. If the total distance to be travelled by 200 students is to be as small as possible, then the school should be built at (a)town B (b)town A (c)45 km from town A (d)45 km from town B  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum Assuming the petrol burnt (per hour) in driving a motor boat veries as the cube of its velocity, show that the most economical speed when going against the current of c 203 miles per hour is

( ) 3c 2

miles per hour.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 204

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum A point on the hypotenuse of a triangle is at distance aandb from the sides of the 205

( ) 3

2

triangle. Show that the minimum length of the hypotenuse is a 2 + b 3 .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 206

Rectangles are inscribed inside a semi-circle of radius r. Find the rectangle with maximum area.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 207

A running track of 440 ft is to be laid out enclosing a football field, the shape of which is a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum, then find the length of its sides.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

208

If the sum of the lengths of the hypotenuse and another side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between these π sides is .

3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

209

The tangent to the parabola y = x2 has been drawn so that the abscissa x 0 of the point of tangency belongs to the interval [1,2]. Find x 0 for which the triangle bounded by the tangent, the axis of ordinates, and the straight line y = x02 has the greatest area.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 210

LL′ is the latus sectum of the parabola y 2 = 4axandPP′ is a double ordinate drawn between the vertex and the latus rectum. Show that the area of the trapezium PP ′ LL′ is maximum when the distance PP′ from the vertex is a /9.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 211

Find the points on the curve 5x 2 - 8xy + 5y 2 = 4 whose distance from the origin is maximum or minimum.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry 212

OF

DERIVATIVES_Solid

A sheet of area 40m 2 is used to make an open tank with square base. Find the dimensions of the base such that the volume of this tank is maximum.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry 213

OF

DERIVATIVES_Solid

The latest edge of a regular hexagonal pyramid is 1cm. If the volume is maximum, then find its height.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry 214

OF

DERIVATIVES_Solid

Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

215

OF

DERIVATIVES_Solid

Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi-vertical angle α is one-third that of the cone and the 4 greatest volume of cylinder is πh 2 tan 2 α.

27  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

216

OF

DERIVATIVES_Solid

A private telephone company serving a small community makes a profit of Rs. 12.00 per subscriber, if it has 725 subscribers. It decides to reduce the rate by a fixed sum for each subscriber over 725, thereby reducing the profit by 1 paise per subscriber.

Thus, there will be profit of Rs. 11.99 on each of the 726 subscribers, Rs. 11.98 on each of the 727 subscribers, etc. What is the number of subscribers which will give the company the maximum profit?  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry 217

OF

DERIVATIVES_Solid

The lateral edge of a regular rectangular pyramid is acmlong. The lateral edge makes an angle α with the plane of the base. Find the value of α for which the volume of the pyramid is greatest.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry 218

OF

DERIVATIVES_Solid

The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

219

OF

DERIVATIVES_Solid

A figure is bounded by the curves y = x 2 + 1, y = 0, x = 0, andx = 1. At what point (a, b) should a tangent be drawn to curve y = x 2 + 1 for it to cut off a trapezium of greatest area from the figure?  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry 220

OF a2

DERIVATIVES_Solid

b2

Find the minimum length of radius vector of the curve 2 + 2 = 1 x y  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry 221

OF

DERIVATIVES_Solid

Prove that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to 2/3 rd the diameter of the sphere.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry 222

OF

DERIVATIVES_Solid

Find the point at which the slope of the tangent of the function f(x) = excosx attains minima, when x ∈ [0, 2π].  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

223

OF

DERIVATIVES_Solid

An electric light is placed directly over the centre of a circular plot of lawn 100 m in diameter. Assuming that the intensity of light varies directly as the sine of the angle at which it strikes an illuminated surface and inversely as the square of its distance from its surface, how should the light be hung in order that the intensity may be as great as possible at the circumference of the plot?  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

OF

DERIVATIVES_Solid

224

A regular square based pyramid is inscribed in a sphere of given radius R so that all vertices of the pyramid belong to the sphere. Find the greatest value of the volume of the pyramid.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 2

225

2

2

Prove that the equation of the normal to x 3 + y 3 = a 3 is ycosθ - xsinθ = acos2θ, where θ is the angle which the normal makes with the axis of x.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal

226

Show that the segment of the tangent to the curve y =

a 2

In

(

√a 2 - x2 a - √a 2 - x 2

a+

)

-

√a 2 - x 2

contained between the y=axis and the point of tangency has a constant length.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

227

(

)

(

)

If the tangent at x1 , y1 to the curve x 3 + y 3 = a 3 meets the curve again in x 2, y 2 , x2 y2 then prove that + = -1 x1 y1  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

228

If the area of the triangle included between the axes and any tangent to the curve x ny = a n is constant, then find the value of n.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves

229

OF

DERIVATIVES_Angle

Show the condition that the curves ax 2 + by 2 = 1 and a ′ x 2 + b ′ y 2 = 1 should intersect 1 1 1 a orthogonally is - = ′ - ′ . a b a b  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 230

OF

DERIVATIVES_Angle

Find the angle of intersection of curves y = [|sinx| + |cosx|]andx 2 + y 2 = 5, where [.] denotes the greatest integral function.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

231

Tangents are drawn from the origin to curve y = sinx. Prove that points of contact lie x2 2 on y = 1 + x2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 232

OF

Prove that the curves y = f(x), [f(x) > 0], andy = f(x)sinx, wheref(x) is differentiable function, have common tangents at common points.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

233

Sand is pouring from a pipe at the rate of 12cm 3 /s. The falling sand forms a cone on the ground in such a way that the height of the cone is always 1/6th of the radius of the base. How fast does the height of the sand cone increase when the height in 4 cm?  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

234

a0

Let

a1

a2

an - 1

+ + + + + an = 0. Show that there exists at least real x n+1 n n-1 2 between 0 and 1 such that a 0x n + a 1 xn - 1 + a 2x n - 2 + + an = 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems Let a, b, c be nonzero real numbers such that 235

2

(

)(

∫

1 0

(1 + cos x )(ax 8

2

)

+ bx + c dx

)

= ∫ 0 1 + cos 8x ax 2 + bx + c dx = 0 Then show that the equation ax 2 + bx + c = 0 will have one root between 0 and 1 and other root between 1 and 2.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 236

If f is continuous and differentiable function and f(0) = 1, f(1) = 2, then prove that there exists at least one c ∈ [0, 1]f or whichf ′ (c)(f(c)) n - 1 > 2 n - 1 , where n ∈ N.

√

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous 237

OF

Let a, b, c be three real numbers such that `a  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal

238

x+ Prove that the portion of the tangent to the curve

√a 2 - y 2

= (log) e a intercepted between the point of contact and the x-axis is constant.

a+

√a 2 - y 2 y

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 239

OF

DERIVATIVES_Angle

Find the condition for the line y = mx to cut at right angles the conic ax 2 + 2hxy + by 2 = 1.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal 240

Show that for the curve by 2 = (x + a) 3 , the square of the sub-tangent varies as the sub-normal.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure

241

2 An aeroplane is flying horizontally at a height of km with a velocity of 15 km/h. Find 3 the rate at which it is receding from a fixed point on the ground which it passed over 2 min ago.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 242

Use the mean value theorem to prove e x ≥ 1 + x ∀ x ∈ R  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications

243

OF

Show that the square roots of two successive natural numbers greater than N2 differ 1 by less than .

2N  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 3

244

3

3

The number of tangents to the curve x 2 + y 2 = 2a 2 , a > 0, which are equally inclined to the axes, is 2 (b) 1 (c) 0 (d) 4  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

245

The angle made by the tangent of the curve x = a(t + s ∈ tcost), y = a(1 + s ∈ t) 2 with 1 1 - sint 1 1 + sint the x-axis at any point on it is (a) (π + 2t) (b) (c) (2t - π) (d) 4 cost 4 cos2t  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 246

If m is the slope of a tangent to the curve e y = 1 + x 2 , then |m| > 1 (b) m > 1 m ≻ 1 (d) |m| ≤ 1  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

247

If at each point of the curve y = x 3 - ax 2 + x + 1, the tangent is inclined at an acute angle with the positive direction of the x-axis, then a > 0 (b) a < - √3 - √3 ≤ a ≤ √3 (d) noneofthese  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 248

The slope of the tangent to the curve y =

√4 - x 2 at the point where the ordinate and

the abscissa are equal is - 1 (b) 1 (c) 0 (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 249

The curve given by x + y = e xy has a tangent parallel to the y - aξs at the point (0, 1) (b) (1, 0) (1, 1) (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

250

If the line joining the points (0, 3)and(5, - 2) is a tangent to the curve y =

C x+1

, then

the value of c is 1 (b) - 2 (c) 4 (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 251

Let f(x) =

{ - x , f or x < 0x 2

2

+ 8, f or x ≥ 0thenx - ∈ tercep → fthe line, that is, the

tangent to the graph of f(x) , is zero (b) - 1 (c) - 2 (d) - 4

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 252

The distance between the origin and the tangent to the curve y = e 2x + x 2 drawn at the point x = 0 is  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals The normal to the curve 2x 2 + y 2 = 12 at the point (2, 2) cuts the curve again at 253

(

-

22 9

, -

2 9

) ( ) (b)

22 2 , ( - 2, - 2) (d) none of these 9 9

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

254

2 3 1 2 At what point of curve y = x + x , the tangent makes equal angle with the axis?

3 2

( ) (

)( )

( ) ( )( ) (

1 5 1 1 4 1 1 1 1 4 1 , and - 1, , and( - 1, 0) , and - 3, , and - 1, 5 24 6 2 9 3 7 2 3 47 3

)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals The equation of the tangent to the curve y = be - x/ a at the point where it crosses the y255

axis is a

)

x a

-

y

x y = 1 (b) ax + by = 1 c)ax - by = 1 (d) + = 1 b a b

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves

OF

Then angle of intersection of the normal at the point 256

(

DERIVATIVES_Angle

-

5

√2

,

3

√2

)

of the curves

π π π x 2 - y 2 = 8 and 9x 2 + 25y 2 = 225 is 0 (b) (c) (d) 2 3 4  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 257

OF

A function y - f(x) has a second-order derivative f x = 6(x - 1) . It its graph passes through the point (2,1) and at that point tangent to the graph is y = 3x - 5, then the value of f(0) is 1 (b) - 1 (c) 2 (d) 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 258

If x + 4y = 14 is a normal to the curve y 2 = αx 3 - β at (2,3), then the value of α + β is 9 (b) - 5 (c) 7 (d) - 7  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals In the curve represented parametrically by the equations x = 21ncott + 1andy = tant + cott, A. tangent and normal intersect at the point (2,1) B.

259

normal at t =

π 4

is parallel to the y-axis. C. tangent at t =

D. tangent at t =

π 4

π 4

is parallel to the line y = x

is parallel to the x-axis.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals The abscissas of point PandQ on the curve y = e x + e - x such that tangents at PandQ make 260 1n

(

600

√7 - √3 2

with the x-axis are. 1n

) ( ±1n

√3 + √7 2

(

√3 + √7 7

) ( and1n

√3 + √5 2

) ( 1n

√3 + √7 2

)

(c)

)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

261

If a variable tangent to the curve x 2y = c 3 makes intercepts a, bonx - andy - axes, 4 3 27 3 4 3 2 3 respectively, then the value of a b is 27c (b) c (c) c (d) c 27 4 9  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

262

Let C be the curve y = x 3 (where x takes all real values). The tangent at A meets the curve again at B. If the gradient at B is K times the gradient at A, then K is equal to 4 1 (b) 2 (c) - 2 (d) 4  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal

263

A curve is represented by the equations x = sec 2 tandy = cott, where t is a parameter. If π the tangent at the point P on the curve where t = meets the curve again at the point 4 5√3 5√ 5 2√5 3√ 5 Q, then |PQ| is equal to (b) (c) (d) 2 2 3 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

264

a b The x-intercept of the tangent at any arbitrary point of the curve 2 + 2 = 1 is x y proportional to square of the abscissa of the point of tangency square root of the abscissa of the point of tangency cube of the abscissa of the point of tangency cube root of the abscissa of the point of tangency  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal

265

At any point on the curve 2x 2 y 2 - x 4 = c, the mean proportional between the abscissa and the difference between the abscissa and the sub-normal drawn to the curve at the same point is equal to or d ∈ ate (b) radius vector x - ∈ tercep → ftan ≥ nt (d) subtangent  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal

266

If the length of sub-normal is equal to the length of sub-tangent at any point (3,4) on the curve y = f(x) and the tangent at (3,4) to y = f(x) meets the coordinate axes at AandB , then the maximum area of the triangle OAB, where O is origin, is 45/2 (b) 49/2 (c) 25/2 (d) 81/2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

267

The number of point in the rectangle {(x, y)} - 12 ≤ x ≤ 12and - 3 ≤ y ≤ 3} which lie on the curve y = x + sinx and at which in the tangent to the curve is parallel to the x-axis is 0 (b) 2 (c) 4 (d) 8  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications

|

OF

|

Tangent of acute angle between the curves y = x 2 - 1 and y = 268

√7 - x 2 at their points

5√3

3√5 5√3 3√5 of intersection is (b)

(d) 2 2 4 4  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves

269

OF

DERIVATIVES_Angle

The line tangent to the curves y 3 - x 2 y + 5y - 2x = 0 and x 2 - x 3y 2 + 5x + 2y = 0 at the π π π π origin intersect at an angle θ equal to (b) (c) (d) 6 4 3 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves

270

OF

DERIVATIVES_Angle

1 The two curves x = y2 , xy = a 3 cut orthogonally at a point. Then a 2 is equal to (b) 3 3 1 (c) 2 (d) 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION

OF

DERIVATIVES_Angle

Between The Curves 271

The curves 4x2 + 9y 2 = 72 and x 2 - y 2 = 5at(3, 2) touch each other (b) cut orthogonally intersect at 450 (d) intersect at 60 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 272

OF

Let f(1) = - 2andf ′ (x) ≥ 4. 2f or 1 ≤ x ≤ 6. The smallest possible value of f(6) is 9 (b) 12 (c) 15 (d) 19  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 273

If f(x) = x 3 + 7x - 1, then f(x) has a zero between x = 0andx = 1 . The theorem that best describes this is mean value theorem maximum-minimum value theorem intermediate value theorem none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications

274

Consider the function f(x) =

{

x

sinπ x

OF

, f or x > 00, f or x = 0 The, the number of point in

(0,1) where the derivative f ′ (x) vanishes is 0 (b) 1 (c) 2 (d) infinite  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 275

OF

Let f(x)andg(x) be differentiable for 0 ≤ x ≤ 1, such that f(0) = 0, g(0) = 0, f(1) = 6. Let there exists real number c in (0,1) such taht f ′ (c) = 2g ′ (c) . Then the value of g(1) must be 1 (b) 3 (c) - 2 (d) - 1  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 276

If 3(a + 2c) = 4(b + 3d), then the equation ax3 + bx 2 + cx + d = 0 will have no real solution at least one real root in ( - 1, 0) at least one real root in (0, 1) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

277

If the function f(x) = ax 3 + bx 2 + 11x - 6 satisfies conditions of Rolles theorem in 1 [1, 3]f or x = 2 + , then values of aandb , respectively, are - 3, 2 (b) 2, - 4 1, 6 (d) √3 none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

278

A value of C for which the conclusion of mean value theorem bolds for the function 1 f(x) = glo ex on the interval [1, 3] is (log) e3 (b) (log) 3e (log) e3 (d) 2(log) 3e 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

279

Let f(x) be a twice differentiable function for all real values of x and satisfies f(1) = 1, f(2) = 4, f(3) = 9. Then which of the following is definitely true? f x = 2 ∀ x ∈ (1, 3) f x = f(x) = 5f or somex ∈ (2, 3) f x = 3 ∀ x ∈ (2, 3) f x = 2f or somex ∈ (1, 3)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems The value of c in Lagranges theorem for the function f(x) = logsinx in the interval 280

[ ]

π 5π π π 2π , is (b) (d) none of these 6 6 4 2 3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems In

which

of

the

following

functions

(a)f(x) = {x, 0 ≤ x < 10, x = 1on[0, 1] 281 (c)f(x) =

x2 - x - 6 x-1

on[ - 2, 3] (d)f(x) =

{

is

(b)f(x) =

Rolles

{

x 3 - 2x 2 - 5x + 6 x-1

sinx x

theorem

applicable?

, - π ≤ x < 00, x = 0on[ - π, 0)

if x ≠ 1, - 6 if x = 1on[ - 2, 3]

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure A point on the parabola y 2 = 18x at which the ordinate increases at twice the rate of 282

the abscissa is (a)(2,6) (b) (2, - 6) (c)

( ) ( ) 9 8

, -

9 2

(d)

9 9 , 8 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation

Of Dy/Dx As A Rate Measure 283

The rate of change of the volume of a sphere w.r.t. its surface area, when the radius is 2 cm, is 1 (b) 2 (c) 3 (d) 4  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Approximations

284

OF

If there is an error of k % in measuring the edge of a cube, then the percent error in k estimating its volume is (a)k (b) 3k (c) (d) none of these 3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure

285

A lamp of negligible height is placed on the ground l 1 away from a wall. A man l 2m tall l1 is walking at a speed of m/s from the lamp to the nearest point on the wall. When 10 he is midway between the lamp and the wall, the rate of change in the length of this 5l 2 2l 2 l2 l2 shadow on the wall is m /s (b) m /s - m /s (d) - m /s 2 5 2 5  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure

286

A man is moving away from a tower 41.6 m high at the rate of 2 m/sec. Find the rate at which the angle of elevation of the top of tower is changing, when he is at a distance of 30m from the foot of the tower. Assume that the eye level of the man is 1.6m from the ground.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 287

OF

The coordinates of a point on the parabola y 2 = 8x whose distance from the circle x 2 + (y + 6) 2 = 1 is minimum is (a)(2, 4) (b) (2, - 4) (c)(18, - 12) (d) (8, 8)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications

OF

( )

At the point P a, a n on the graph of y = xn , (n ∈ N), in the first quadrant, a normal is 288

drawn. The normal intersects the y - aξs at the point (0, b). If (lim) a 0b =

1

→

equals 1 (b) 3 (c) 2 (d) 4

2

, then n

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure 289

Suppose that f is differentiable for all x and that f ′ (x) ≤ 2f or allx. If f(1) = 2andf(4) = 8, thenf(2) has the value equal to 3 (b) 4 (c) 6 (d) 8  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure

290

The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height decreases at the rate of 0.2 cm/min. The rate of change of the volume of the 8π 3π cylinder, in cm 2 /m ∈ , when the radius is 2cm and the height is 3cm is - 2p (b) - 5 5 2π (d) 5  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure A cube of ice melts without changing its shape at the uniform rate of 4 291

of change of the surface area of the cube, in

cm 2 m∈

cm 3 m∈

. The rate

, when the volume of the cube is

125cm 3 , is - 4 (b) -

16 5

(c) -

16 6

(d) -

8 15

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems The tangent to the curve y = e kx at a point (0,1) meets the x-axis at (a,0), where 292

[ ] [

a ∈ [ - 2, - 1] . Then k ∈ -

1 2

, 0 (b) - 1, -

1 2

]

[0, 1] (d)

[ ] 1 2

,1

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous 293

OF

Let `f^(prime)(x)=e^x^2a n df(0)=10.IfA  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

294

If f is a continuous function on [0, 1], differentiable in (0, 1) such that f(1) = 0, then there exists some c ∈ (0, 1) such that cf ′ (c) - f(c) = 0 cf ′ (c) + cf(c) = 0 f ′ (c) - cf(c) = 0 cf ′ (c) + f(c) = 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

295

Given g(x) =

x+2

and the line 3x + y - 10 = 0. Then the line is tangent to g(x) (b) x-1 normal to g(x) chord of g(x) (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 296

Let f be a continuous, differentiable, and bijective function. If the tangent to y = f(x)atx = a is also the normal to y = f(x)atx = b, then there exists at least one c ∈ (a, b) such that f ′ (c) = 0 (b) f ′ (c) > 0 f ′ (c) < 0 (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

297

If f(x)andg(x) are differentiable functions for 0 ≤ x ≤ 1 such that f(0) = 10, g(0) = 2, f(1) = 2, g(1) = 4, then in the interval (0, 1). f ′ (x) = 0f or allx f ′ (x) + 4g ′ (x) = 0 for at least one x f(x) = 2g′ (x) for at most one x none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous 298

OF

A continuous and differentiable function y = f(x) is such that its graph cuts line y = mx + c at n distinct points. Then the minimum number of points at which f x = 0 is/are n - 1 (b) n - 3 n - 2 (d) cannot say  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

299

If f(x) is continuous in [a, b] and differentiable in (a,b), then prove that there exists at f ′ (c) f(b) - f(a) least one c ∈ (a, b) such that 2 = 3c b3 - a3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure

300

The radius of the base of a cone is increasing at the rate of 3 cm/min and the altitude is decreasing at the rate of 4 cm/min. The rate of change of lateral surface when the radius is 7 cm and altitude is 24cm is 108πcm 2 / min (b) 7πcm 2 / min 27πcm 2 / min (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals Points on the curve f(x) = 301 x-axis are (0,0) (b)

(

√3,

-

x

π

1 - x2

√3 2

where the tangent is inclined at an angle of to the 4

)( ) ( - 2,

2 3

(d) - √ 3,

√3 2

)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal x

In the curve y = ce a , the sub-tangent is constant sub-normal varies as the square of 302

(

)

the ordinate tangent at x 1 , y 1 on the curve intersects the x-axis at a distance of

(x 1 - a ) from the origin equation of the normal at the point where the curve cuts y - aξs is cy + ax = c 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 303

Let f(x) = a 5 x 5 + a 4x 4 + a 3x 3 + a 2 x2 + a1 x, where a i′ s are real and f(x) = 0 has a positive root α 0. Then f ′ (x) = 0 has a positive root α 1 such that `0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 304

Let the parabolas y = x(c - x)andy = x 2 + ax + b touch each other at the point (1,0).

Then a + b + c = 0 a + b = 2 b - c = 1 (d) a + c = - 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 305

OF

DERIVATIVES_Angle

Which of the following pair(s) of curves is/are orthogonal? y 2 = 4ax; x 2 = 4ayat(0, 0) xy = a 2 ; x2 - y 2 = b 2 y = ax; x2 + y2 = c2

y2

x

= 4ax; y =

e - 2a

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals The coordinates of the point(s) on the graph of the function f(x) = 306

x3

5x 2

+ 7x - 4 , 3 2 where the tangent drawn cuts off intercepts from the coordinate axes which are equal

( ) ( )( )

in magnitude but opposite in sign, are 2,

8 3

(b) 3,

7 2

1,

5 6

(d) none of these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

307

The abscissa of a point on the curve xy = (a + x) 2 , the normal which cuts off 1 a numerically equal intercepts from the coordinate axes, is (b) √2a (c) (d) - √2a √2 √2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves

OF

DERIVATIVES_Angle

( )

5 3 The angle formed by the positive y - aξs and the tangent to y = x 2 + 4x - 17at , 2 4

308

is tan - 1 (9) (b)

π 2

-

tan - 1 (9)

π 2

+ tan - 1 (9) (d) none of these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves

(

OF

DERIVATIVES_Angle

)

If the tangent at any point P 4m 2 , 8m 3 of x3 - y 2 = 0 is also a normal to the curve 309 x3

- y2

= 0, then the value of m is

√2 3

(b) -

√2 3

(c)

3

√2

(d) -

3

√2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves

310

OF

DERIVATIVES_Angle

()

4 The angle between the tangents to the curves y = x2 andx = y 2at(1, 1) is cos - 1 (b) 5 sin - 1

() () 3 5

tan - 1

3 4

(d) tan - 1

() 1 3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves

OF

DERIVATIVES_Angle

Show that the angle between the tangent at any point P and the line joining P to the 311

()

y origin O is same at all points on the curve log x 2 + y 2 = ktan - 1 x

(

 Watch Free Video Solution on Doubtnut

)

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

Given f(x) = 4 312

( ) 1

-x

2 3

, g(x) =

{

tan[x]

, x ≠ 01, x = 0 h(x) = {x}, k(x) = 5 ( log ) 2 ( x + 3 )

2 x Then in [0,1], lagranges mean value theorem is not applicable to (where [.] and {.} represents the greatest integer functions and fractional part functions, respectively). f (b) g (c) k (d) h  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous

313

OF

Which of the following is/are correct? Between any two roots of excosx = 1, there exists at least one root of tanx = 1. Between any two roots of e xsinx = 1, there exists at least one root of tanx = - 1. Between any two roots of e xcosx = 1, there exists at least one root of e xsinx = 1. Between any two roots of e xsinx = 1, there exists at least one root of e xcosx = 1.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves 314

OF

DERIVATIVES_Angle

Which of the following pairs(s) of curves is/are orthogonal? y 2

x

= 4ax; y =

e - 2a y^2 =

4ax; x^2 = 4ay at (0,0) xy = a 2 ; x 2 - y 2 = b 2 y = ax; x 2 + y 2 = c 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Length Of Tangent Normal Sub-Tangent And Sub-Normal

315

The abscissa of the point on the curve √xy = a + x the tangent at which cuts off equal a intercepts from the coordinate axes is (b) a/ √2 (c) - a√2 (d) a√2 √2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

316

Each question has four choices, a,b,c and d, out of which only one is correct. Each question contains STATEMENT 1 and STATEMENT 2. If both the statement are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1. If both the statements are TRUE but STATEMENT 2 is NOT the correct explanation of STATEMENT 1. If STATEMENT 1 is TRUE and STATEMENT 2 is FLASE. If STATEMENT 1 is FALSE and STATEMENT 2 is TURE. Statement 1: Lagrange mean value theorem is not applicable to f(x) = |x - 1|(x - 1) Statement 2: |x - 1| is not differentiable at x = 1.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

317

Statement 1: If 27a + 9b + 3c + d = 0, then the equation f(x) = 4ax 3 + 3bx 2 + 2cx + d = 0 has at least one real root lying between (0, 3) . Statement 2: If f(x) is continuous in [a,b], derivable in (a, b) such that f(a) = f(b), then there exists at least one point c ∈ (a, b) such that f ′ (c) = 0.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

318

Statement 1: If both functions f(t)andg(t) are continuous on the closed interval [1,b], differentiable on the open interval (a,b) and g ′ (t) is not zero on that open interval, then f ′ (c) f(b) - f(a) there exists some c in (a, b) such that ′ = Statement 2: If f(t)andg(t) are g(b) g(a) g (c) continuou and differentiable in [a, b], then there exists some c in (a,b) such that f(b) - f(a) g(b) - g(a) f ′ (c) = andg ′ (c) from Lagranes mean value theorem. b -a b-a  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications Statement 319

(√

- 3 + 4x - x 2 + 4

1:

)

2

The

maximum

OF value

of

+ (x - 5) 2 (where1 ≤ x ≤ 3)is36. Statement 2: The maximum

distance between the point (5, - 4) and the point on the circle (x - 2) 2 + y 2 = 1 is 6  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

320

Statement 1: If g(x) is a differentiable function, g(2) ≠ 0, g( - 2) ≠ 0, and Rolles theorem x2 - 4 is not applicable to f(x) = ∈ [ - 2, 2], theng(x) has at least one root in ( - 2, 2). g(x) Statement 2: If f(a) = f(b), theng(x) has at least one root in ( - 2, 2) . Statement 2: If f(a) = f(b), then Rolles theorem is applicable for x ∈ (a, b) .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 321

Statement 1: The tangent at x = 1 to the curve y = x3 - x 2 - x + 2 again meets the curve at x = 0. Statement 2: When the equation of a tangent is solved with the given curve, repeated roots are obtained at point of tangency.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

322

Statement 1: If f(x) is differentiable in [0, 1] such that f(0) = f(1) = 0, then for any λ ∈ R, there exists c such that f ′ (c) = λf(c),0 < c < 1. statement 2: if g(x) is differentiable in [0,1], where g(0) = g(1), then there exists c such that g ′ (c)=0,  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

323

Statement 1: For the function f(x) = x 2 + 3x + 2, LMVT is applicable in [1, 2] and the value of c is 3 /2 . Statement 2: If LMVT is known to be applicable for any quadratic a+b polynomial in [a, b] , then c of LMVT is .

2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value

Theorems 324

Let y = f(x) be a polynomial of odd degree ( ≥ 3) with real coefficients and (a, b) be any point. Statement 1: There always exists a line passing through (a, b) and touching the curve y = f(x) at some point. Statement 2: A polynomial of odd degree with real coefficients has at least one real root.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

325

There is a point (p,q) on the graph of f(x) = x 2 and a point (r, s) on the graph of -8 g(x) = , wherep > 0andr > 0. If the line through (p, q)and(r, s) is also tangent to both x the curves at these points, respectively, then the value of P + r is_________.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves

326

OF

DERIVATIVES_Angle

A curve is defined parametrically be equations x = t 2 andy = t 3 . A variable pair of perpendicular lines through the origin O meet the curve of PandQ . If the locus of the point of intersection of the tangents at PandQ is ay 2 = bx - 1, then the value of (a + b) is____  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 327

OF

If d is the minimum distance between the curves f(x) = exandg(x) = (log) ex, then the value of d 6 is  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications

328

OF

Let f(x0 be a non-constant thrice differentiable function defined on ( - ∞, ∞) such that f(x) = f(6 - x)andf ′ (0) = 0 = f ′ (x) 2 = f(5) . If n is the minimum number of roots of n f ′ (x) 2 + f ′ (x)f x = 0 in the interval [0,6], then the value of is___ 2

(

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications

OF

( )

At the point P a, a n on the graph of y = xn , (n ∈ N) , in the first quadrant, a normal is 329

drawn. The normal intersects the y - aξs at the point (0, b) . If (lim) a0 =

1

→

2

, then n

equals _____.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Between The Curves

330

OF

DERIVATIVES_Angle

A curve is given by the equations x = sec 2 θ, y = cotθ. If the tangent at Pwhereθ =

π

4 meets the curve again at Q, then[PQ] is, where [.] represents the greatest integer function, _________.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Interpretation Of Dy/Dx As A Rate Measure 331

Water is dropped at the rate of 2 m 3/s into a cone of semi-vertical angle is 45 ∘ . If the rate at which periphery of water surface changes when the height of the water in the cone is 2m is d. Then the value of 5d is _____ m/sec  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 332

If the slope of line through the origin which is tangent to the curve y = x 3 + x + 16 is m, then the value of m - 4 is____.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals Let y=f(x) be drawn with f(0) =2 and for each real number a the line tangent to y = f(x)

333

at (a,

f(a)) has x-intercept (a-2). If f(x) is of the form of ke px then

to

k p

has the value equal

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 334

Suppose a, b, c are such that the curve y = ax 2 + bx + c is tangent to y = 3x - 3at(1, 0) and is also tangent to y = x + 1at(3, 4). Then the value of (2a - b - 4c) equals ______  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 335

Let C be a curve defined by y = e a + bx 2. The curve C passes through the point P(1, 1) and the slope of the tangent at P is ( - 2). Then the value of 2a - 3b is_____.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 336

OF

If the curve C in the xy plane has the equation x 2 + xy + y 2 = 1, then the fourth power of the greatest distance of a point on C from the origin is___.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications

337

OF

If a, b are two real numbers with a < b , then a real number c can be found between a 2 + ab + b 2 a and b such that the value of is _ _ _ 2 c  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 338

Let f defined on [0, 1] be twice differentiable such that |f(x)| ≤ 1 for x ∈ [0, 1]. if f(0) = f(1) then show that ∣f′ (x) < 1 for all x ∈ [0, 1].  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 339

Let fandg be differentiable on [0,1] such that f(0) = 2, g(0), f(1) = 6andg(1) = 2. Show that there exists c ∈ (0, 1) such that f ′ (c) = 2g ′ (c) .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 340

OF

Find the shortest distance of the point (0, c) from the parabola y = x 2 , where 0 ≤ c ≤ 5 .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 341

Find the equation of tangents to the curve y = cos(x + y), - 2π ≤ x ≤ 2π that are parallel to the line x + 2y = 0.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 342

(

)

Find the equation of the normal to the curve y = (1 + y) y + sin - 1 sin 2x atx = 0.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 343

The curve y = ax 3 + bx 2 + cx + 5 touches the x-axis at P( - 2, 0) and cuts the y-axis at the point Q where its gradient is 3. Find the equation of the curve completely.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems

344

(

1

)

Using Rolles theorem, prove that there is at least one root in 45 100 , 46 of the equation. P(x) = 51x 101 - 2323(x) 100 - 45x + 1035 = 0.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 345

OF

| ( ) ( )| ≤ (x1 - x2 )2Find the equation of gent to the curve y = f(x) at the point

if f x 1 - f x 2 (1, 2).

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 346

OF

If H is the number of horizontal tangents and V is the number of vertical tangents to the curve y 3 - 3xy + 2 = 0, then the value of (H + V) equals  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems 347

If a, b, c ∈ R and a + b + c = 0, then the quadratic equation 3ax 2 + 2bx + c = 0 has (a) at least one root in [0, 1] (b) at least one root in [1,2] (c) at least one root in [3/2, 2] (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 348

Show that the normal at any point θ to the curve x = acosθ + aθsinθ, y = asinθ - a θcosθ is at a constant distance from the origin.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 349

The slope of the tangent to a curve y = f(x) at (x, f(x))is2x + 1. If the curve passes through the point (1,2) then the area of the region bounded by the curve, the x-axis and the line x=1 is  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

350

If the normal to the curve y = f(x) at the point (3, 4) makes an angle

3π 4

with the

positive x-axis, then f′ (3) =  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals 351

The triangle formed by the tangent to the curve f(x) = x2 + bx - b at the point (1, 1) and the coordinate axes, lies in the first quadrant. If its area is 2, then the value of b is  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Mean Value Theorems In

[0, 1]

Lagranges

Mean

Value

theorem

in

NOT

applicable

to

352 f(x) =

{

1 2

- × <

( )

1 1 2 2

-x

2

x≥

1 2

b. f(x) =

{

s∈x x

, x ≠ 01, x = 0 c. f(x) = x|x| d. f(x) = |x|

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Equation Of Tangents And Normals

( )

The tangent to the curve y = e x drawn at the point c, e c intersects the line joining 353

(c - 1, e ) and (c + 1, e ) c- 1

c +1

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 354

OF

If the line ax + by + c = 0 is a normal to the curve xy = 1, then a > 0, b > 0 a > 0, b < 0 a〈0, b〉0 (d) a < 0, b < 0 none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous Applications 355

OF

Which one of the following curves cut the parabola at right angles? x 2 + y 2 = a 2 (b) y = e - x/ 2a y = ax (d) x 2 = 4ay  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION

OF

DERIVATIVES_Monotonocity : Introductions 356

Find the values of x where f(x) = sin(lnx) - cos(lnx) in strictly increasing.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 357

OF

Leg f(x) = x 3 + ax 2 + bx + 5sin 2x be an increasing function on the set R. Then find the condition on aandb.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 358

OF

Find the possible values of a such that f(x) = e 2x - (a + 1)ex + 2x is monotonically increasing for x ∈ R.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 359

Prove that for any two numbers x1 andx 2.

e 2x 1 + e x 2 3

>e

2x 1 + x 2 3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 360

OF

If `  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION

OF

DERIVATIVES_Monotonocity : Introductions

361

Discuss the monotonocity of Q(x), where Q(x) = 2f

() x2 2

(

)

+ f 6 - x2 ∀ x ∈ R It is given

that f x > 0 ∀ x ∈ R. Find also the point of maxima and minima of Q(x) .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

362

Prove that

( )

tan - 1 1 2 e

+

OF

2e

(

) (

e 2 + 1 < tan - 1 e

)

2

+

2

√ e2 + 1

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 363

Prove that `sin^2theta  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 364

OF

OF

Let f(x) = x 3 - 3x 2 + 6 ∀ x ∈ R and g(x) = {max : f(t; x + 1 ≤ t ≤ x + 2, - 3 ≤ x ≤ 01 - x, x ≥  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

365

OF

If f is a real function such that f(x) > 0, f ′ ( × ) is continuous for all real √f(1) ′ xandaxf (x) ≥ 2√ f(x) - 2af(x), (ax ≠ 2), show that √f(x) ≥ , x ≥ 1. x  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous 366

OF

The lower corner of a leaf in a book is folded over so as to reach the inner edge of the page. Show that the fraction of the width folded over when the area of the folded part is minimum is 2 /3.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 367

From a fixed point A on the circumference of a circle of radius r, the perpendicular AY falls on the tangent at P. Find the maximum area of triangle APY.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

368

For what values of a, the function f(x) =

{( ) √a + 4 1-a

x 5 - 3x + log(5) decreases for all

real x.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 369

Find the greatest value of f(x)

1 2

2ax - x - 5a

2

OF

∈ [ - 3, 5] depending upon the parameter a.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous

OF

PandQ are two points on a circle of centre C and radius α . The angle PCQ being 2θ ,

370

find the value of sinθ when the radius of the circle inscribed in the triangle CPQ is maximum.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

371

Of f(x) =

2

tan - 1

( ) 2x + 1

(

- log x 2 + x + 1

)(λ

√3 √3 for all x ∈ R, find the permissible values of λ.

2-

OF

DERIVATIVES_Solid

)

5λ + 3 x + 10 is a decreasing function

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 372

Discuss the number of roots of the equation e(k - xlogx) = 1 for different value of k.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum Prove that sin1 > cos(sin1). Also, show that the equation sin(cos(sinx)) = cos(sin(cosx)) 373

[ ]

has only one solution in 0,

π 2

.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions →

374

OF

Let f: RR be a twice differentiable function such that f(x + π) = f(x)andf x + f(x) ≥ 0 for all x ∈ R. Show that f(x) ≥ 0 for all x ∈ R.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 375

Show that 5x ≤ 8sinx - sin2x ≤ 6xf or 0 ≤ x ≤

π 3

OF

.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

OF

Let f(x), x ≥ 0, be a non-negative continuous function. If f ′ (x)cosx ≤ f(x)sinx ∀ x ≥ 0, 376

then find f

( ) 5π 3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 377

OF

If f(x) = kx 3 - 9x2 + 9x + 3 monotonically increasing in R, then k < 3 (b) k ≤ 2 k ≥ 3 (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

378

If the function f(x) =

Ksinx + 2cosx

sinx + cosx (b) K > 1 K < 2 (d) K > 2

is strictly increasing for all values of x, then K < 1

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 379

OF

Let f: R → R be a function such that f(x) = ax + 3sinx + 4cosx. Then f(x) is invertible if (a) a ∈ ( - 5, 5) (b) a ∈ ( - ∞, 5) (c)a ∈ ( - 5, + ∞) (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

Let g(x) = 2f 380

() x

2

+ f(2 - x)andf ′ ′ (x) < 0 ∀ x ∈ (0, 2). Then g(x) increases in (a)

( ) 4 3

OF

( ) 1 2

, 2 (b)

( )

, 2 (c)(0, 2) (d) 0,

4 3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

OF

( )

On which of the following intervals is the function x 100 + sinx - 1 decreasing? (a) 0, 381 (b) (0, 1) (c)

π 2

( ) π 2

, π (d) none of these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

OF

A function is matched below against an interval where it is supposed to be increasing.

Which of the following parts is incorrectly matched? Interval, Function (a)[2, ∞) , 382

2x 3

- 3x 2

- 12x + 6 (b)(

- ∞, ∞) , x 3

=

3x2

+ 3x + 3 (c)( - ∞ -

3x 2 - 2x + 1  Watch Free Video Solution on Doubtnut

4) , x 3

+

6x 2

( )

+ 6 (d) - ∞,

1 3

,

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

383

OF

( ) ( )

The function f(x) = tan - 1(sinx + cosx) is an increasing function in -

π π π , (b) 0,

2 4 2

( ) ( ) π π π π - , (d) , 2 2 4 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 384

OF

( )

The function x x decreases in the interval (0, e) (b) (0, 1) 0,

1 e

(d) none of these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 5

385

The function f(x) =

5

∑ (x - K) 2 assumes then minimum value of x given by (a) 5 (b) 2

k =1

(c) 3 (d) 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

386

OF

Which of the following statement is always true? (a)If f(x) is increasing, the f - 1 (x) is 1 decreasing. (b)If f(x) is increasing, then is also increasing. (c)If fandg are positive f(x) f functions and f is increasing and g is decreasing, then is a decreasing function. (d)If g

fandg are positive functions and f is decreasing and g is increasing, the decreasing function.

f g

is a

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

387

OF

Let f: R → R be a differentiable function for all values of x and has the property that f(x)andf ′ (x) has opposite signs for all value of x. Then, (a)f(x) is an increasing function (b)f(x) is an decreasing function (c)f 2 (x) is an decreasing function (d)|f(x)| is an increasing function  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 388

OF

Let f(x) = x 4ax - x 2 , (a > 0) . Then f(x) is a. increasing in (0,3a) decreasing in (3a, 4a)

√

b. increasing in (a, 4a) decreasing in(5a, ∞) c. increasing in (0, 4a) d. none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

389

Let f: R → be a differentiable function ∀ x ∈ R . If the tangent drawn to the curve at any point x ∈ (a, b) always lies below the curve, then (a) f ′ (x) < 0, f x < 0 ∀ x ∈ (a, b) (b) f ′ (x) > 0, f x > 0 ∀ x ∈ (a, b) (c)f ′ (x) > 0, f x > 0 ∀ x ∈ (a, b) (d)noneofthese  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum If f ′ (x) = |x| - {x}, where {x} denotes the fractional part of x, then f(x) is decreasing in

390

( ) ( )( ) ( ) -

1 2

, 0 (b) -

1

1 1 , 2 - , 2 (d) , ∞ 2 2 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 391

OF

Function f(x) = |x| - |x - 1| is monotonically increasing when x < 0 (b) x > 1 x < 1 (d) `0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 392

OF

Let f be continuous and differentiable function such that f(x)andf ′ (x) has opposite signs everywhere. The f is increasing f is decreasing |f| is non-monotonic |f| is decreasing  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 393

OF

If the function f(x) increases in the interval (a, b), andφ(x) [f(x)] 2 , then φ(x) increases in (a, b) φ(x) decreases in (a, b) we cannot say that φ(x) increases or decreases in (a, b). none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 394

OF

If φ(x) is a polynomial function and φ ′ (x) > φ(x) ∀ x ≥ 1andφ(1) = 0, then φ(x) ≥ 0 ∀x ≥ 1 `varphi(x)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection Which 395

f(x) =

{

of

the

following x3

√x, x ≥ 1x 3, 0 ≤ x ≤ 1 3

statements

is

true

for

the

function

- 4x, x < 0 It is monotonic increasing ∀ x ∈ R. f ′ (x) fails

to exist for three distinct real values of x. f ′ (x) changes its sign twice as x varies from - ∞ → ∞ . The function attains its extreme values at x 1 andx 2, such that x 1 andx 2 > 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 396

OF

If `f"(x)>0AAx in R ,f'(3)=0,a n dg(x)=f("tan"hat2x-2"tan"x+4),0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

397

OF

[

Let f(x) be a function such that f(x) = (log) 1 (log) 3(sinx + a]. If f(x) is decreasing for all 2 real values of x, then a ∈ (1, 4) (b) a ∈ (4, ∞) a ∈ (2, 3) (d) a ∈ (2, ∞)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous 398

If

f(x) = x + sinx, g(x) = e - x, u =

√c + 1 - √c,

v=

√c

OF - √ c - 1, (c > 1),

then

(a)

fog(u) < fog(v) (b)gof(u) < gof(v)(c) gof(u) > gof(v) (d) fog(u) > fog(v)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

399

OF

The length of the largest continuous interval in which the function f(x) = 4x - tan2x is π π π π monotonic is (b) (c) (d) 2 4 8 16  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

OF

400 f(x) = (x - 1)|(x - 2)(x -

(

2+

1

√3

3)|.

(

Then f decreases in (a) 2 -

1

√3

) (

)

1

, 2 (b) 2, 2 +

√3

(c)

)

, 4 (d) (3, ∞)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 401

The number of solutions of the equation x 2 + 2x 2 + 6x + 2cosx = 0 ∈ [0, 2π] is one (b) two (c) three (d) zero  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

402

OF

( )

f(x) = (x - 2)|x - 3| is monotonically increasing in (a) - ∞,

5 2

∪ (3, ∞) (b)

( ) 5 2

, ∞ (c)

(2, ∞) (d) ( - ∞, 3)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions f(x) = (x - 8) 4 (x - 9) 5 , 0 ≤ x ≤ 10, monotonically decreases in 403 (d)

( ) 76 9

, 10

 Watch Free Video Solution on Doubtnut

OF

( ] ( ) 76

9, 10

(b)

8, 76 9

(0, 8)

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 404

OF

If f(x) = x 3 + 4x 2 + λx + 1 is a monotonically decreasing function of x in the largest

(

possible interval - 2, -

2 3

)

. Then λ

= 4 (b) λ = 2 λ = - 1 (d) λ has no real value

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 405

( ) ( )

( )

1 1 1 f(x) = xlog ex monotonically decreases in (a) 0, (b) , 1 (c)(1, ∞) (d) , ∞ e e e

|

|

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 406

OF

Given that f ′ (x) > g ′ (x) for all real x, andf(0) = g(0) Then `f(x)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions A function g(x) is defined as g(x) =

407

1

4 (

)

f 2x 2 - 1 +

OF 1 2

(

)

f 1 - x2 andf(x) is an increasing

function. Then g(x) is increasing in the interval. ( - 1, 1)

(√ √) -

2 , 3

2 (d) none of these 3

 Watch Free Video Solution on Doubtnut

( √ ) (√ ) -

2 ,0 ∪ 3

2 ,∞ 3

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

(

)

Let f(x) be a function defined as follows: f(x) = sin x 2 - 3x , x ≤ 0; and6x + 5x 2 , x > 0 408

Then at x = 0, f(x) has a local maximum has a local minimum is discontinuous (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 409

(

)

The greatest value of f(x) = cos xe [ x ] + 7x 2 - 3x , x ∈ [ - 1, ∞], is (where [.] represents the greatest integer function). - 1 (b) 1 (c) 0 (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 410

If f(x) = x 5 - 5x 4 + 5x 3 - 10 has local maximum and minimum at x = pandx = q , respectively, then (p, q) ≡ (0, 1) (b) (1,3) (c) (1,0) (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 411

2 1 The maximum value of (logx) / x is 1 (b) e (d) e e  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 412

If a function f(x) has f ′ (a) = 0andf a = 0, then x = a is a maximum for f(x) x = a is a minimum for f(x) it is difficult to say (a)and(b) f(x) is necessarily a constant function.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous 413

2 The minimum value of 2 x - 3 ^ (3 + 27) is 2 27 (b) 2 (c) 1 (d) none of these

 Watch Free Video Solution on Doubtnut

OF

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 414

The number of real roots of the equation e x- 1 + x - 2 = 0 is 1 (b) 2 (c) 3 (d) 4  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum sin2x

The greatest value of the function f(x) =

( ) π

415

( )

on the interval 0,

sin x + 4

π 2

is

1

√2

(b) √ 2

(c) 1 (d) - √2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous

416

(

The function f(x) = 4sin 2x - 1

)( n

)

x 2 - x + 1 , n ∈ N, has a local minimum at x =

OF π

.

6 Then n is any even number n is an odd number n is odd prime number n is any natural number  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For

Local Maximum/Minimum 417

1

1

Greatest value of f(x) = (x + 1) 3 - (x - 1) 3 on [0,1] is 1 (b) 2 (c) 3 (d)

1 3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

418

If the function f(x) = 2x 3 - 9ax 2 + 12x 2 x + 1, wherea > 0, attains its maximum and 1 2 minimum at pandq , respectively, such that p = q, then a equal to 1 (b) 2 (c) (d) 3 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 419

The real number x when added to its inverse given the minimum value of the sum at x equal to 1 (b) - 1 (c) - 2 (d) 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 420

The function f(x) =

x

2 + has a local minimum at x = 2 (b) x = - 2 x = 0 (d) x = 1 2 x

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

( ) ( )

The maximum value of the function f(x) = sin x + 421

π 6

+ cos x +

π 6

in the interval

( ) 0,

π 2

occurs at

π

π π π (b) (c) (d) 12 6 4 3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 422

Let `f(x)={x+2,-|lt=x<0 1,x=0x/2,0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 423

The maximum value of f(x) =

x

1 1 1 1 is (b) (c) (d) 4 3 6 6 1 + 4x + x 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 424

The maximum slope of the curve y = - x 3 + 3x 2 + 9x - 27 is 0 (b) 12 (c) 16 (d) 32  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 425

Let f(x) = cosπx + 10x + 3x 2 + x 3, - 2 ≤ x ≤ 3. The absolute minimum value of f(x) is 0 (b) - 15 (c) 3 - 2π none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

426

2 The minimum value of e 2x - 2x + 1 sin2x is e (b)

1 e

(c) 1 (d) 0

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 427

The maximum value of x 4 e -̂ x2 is e 2 (b) e - 2 (c) 12e - 2 (d) 4e - 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 428

If a 2 x 4 +

b2 y 4

=

c 6,

then the maximum value of xy is (a)

c2

√ab

(b)

c3 ab

c3

√2ab

(d)

c3 2ab

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

429

(

(

)

The global maximum value of f x0 = (log) 10 4x 3 - 12x 2 + 11x - 3 , x ∈ [2, 3], is 3 3 - (log) 103 (b) 1 + (log) 103 (log) 103 (d) (log) 103 2 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 430

The least natural number a for which x + ax - 2 > 2 ∀x ∈ (0, ∞) is 1 (b) 2 (c) 5 (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

431

A function f is defined by f(x) = |x| m|x - 1| n ∀ x ∈ R. The local maximum value of the mn n mn m (mn) function is (m, n ∈ N), 1 (b) m ∩ ^ m (d) m +n (m + n) (m + n) m + n  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 432

f(x) =

{4x - x

3

(

)

+ ln a 2 - 3a + 3 , 0 ≤ x < 3x - 18, x ≥ 3 complete set of values of a such

that f(x) as a local minima at x = 3 is [ - 1, 2] ( - ∞, 1) ∪ (2, ∞) [1, 2] (d) ( - ∞, - 1) ∪ (2, ∞)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 433

Let the function f(x be defined as follows: `f(x)={x^3+x^2-10 x ,-1lt=x<0cosx ,0lt=x  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 434

A differentiable function f(x) has a relative minimum at x = 0. Then the function f = f(x) + ax + b has a relative minimum at x = 0 for all a and allb (b) all b if a = 0 all b > 0 (d) all a > 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 435

If f(x) = 4x 3 - x 2 - 2x + 1 and `g(x)={min"{"f(t):0lt=x")";0lt=xlt=1 3-x ;1  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection The set of value(s) of a for which the function f(x) = 436

ax 3 3

+ (a + 2)x2 + (a - 1)x + 2

possesses a negative point of inflection is ( - ∞, - 2) ∪ (0, ∞) (b)

{ } -

4 5

( - 2, 0) (d)

empty set  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials 437

Suppose that f is a polynomial of degree 3 and that f x ≠ 0 at any of the stationary point. Then f 1)has exactly one stationary point f 2)must have no stationary point f 3)must have exactly two stationary points f has 4)either zero or two stationary points.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum

438

The maximum value of the function f(x) = 2 - 0 . 4 1 (d) 2 0 . 6

(1 + x) 0 . 6 1+x

0.6

in the interval [0, 1] is 2 0 . 4 (b)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

439

OF

(

)

)

→ f: RR, f(x) is differentiable such that f(f(x)) = k x 5 + x , k ≠ 0 . Then f(x) is always

increasing (b) decreasing either increasing or decreasing non-monotonic  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

440

→ x2 + a Consider the function f : ( - ∞, ∞) - ∞, ∞ defined by f(x) = 2 , a > 0, which of the x +a following is not true? maximum value of f is not attained even though f is bounded. f(x) is increasing on (0, ∞) and has minimum at , = 0 f(x) is decreasing on ( - ∞, 0) and has minimum at x = 0. f(x) is increasing on ( - ∞, ∞) and has neither a local maximum nor a local minimum at x = 0.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum →

441

→

If f : RRandg : RR are two functions such that f(x) + f x = - xg(x)f ′ (x)andg(x) > 0 ∀ x ∈ R. Then the function f 2 (x) + f(′ (x)) 2 has a maxima at x = 0 a minima at x = 0 a point of inflexion at x = 0 none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 442

m

Let h(x) = x n for x ∈ R, where +m and n are odd numbers and `0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

443

f(x) = 4tanx - tan 2x + tan 3 x, x ≠ nπ +

π

, (a)is monotonically increasing 2 monotonically decreasing (c)has a point of maxima (d)has a point of minima  Watch Free Video Solution on Doubtnut

(b)is

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials 444

If for a function f(x), f ′ (a) = 0, f ′ ′ (a) = 0, f a > 0, then at x = a, f(x) is minimum (b) maximum not an extreme point (d) extreme point  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 445

The function f(x) = x(x + 4)e - x/ 2 has its local maxima at x = a. Then (a)a = 2√2 (b) a = 1 - √3 (c)a = - 1 + √ 3 (d) a = - 4  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum

446

{

π si - 1(sinx), x > 0 , x = 0, thencos - 1(cosx), x < 0 x = 0 is a point of maxima x = 0 2 is a point of minima x = 0 is a point of intersection noneofthese Iff(x) =

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum

447

f(x) =

{2 - |x

2

|

+ 5x + 6 , x ≠ 2 and a 2 + 1, x = - 2 Then the range of a, so that f(x) has

maxima at x = - 2, is (a)|a| ≥ 1 (b) |a| < 1 (c)a > 1 (d) a < 1  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum

448

If A > 0, B > 0, andA + B =

π 3

, then the maximum value of tanAtanB is

1

√3

1

(b) (c) 3 (d) 3

√3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum

449

If f(x) =

t + 3x - x 2

, where t is a parameter that has minimum and maximum, then the x -4 range of values of t is (0, 4) (b) (0, ∞) ( - ∞, 4) (d) (4, ∞)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum

450

The value of a for which the function f(x) = asinx +

() 1 3

sin3x has an extremum at x =

π 3

is 1 (b) - 1 (c) 0 (d) 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum The least value of a for which the equation 451

4 sinx

+

1 1 - sinx

= a has at least one solution

( )

in the interval 0,

π 2

(a)9(b) 4 (c)8 (d) 1

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

452

The largest term in the sequence a n = none of these

n2 3

n + 200

is given by (a)

529 49

(b)

8 89

(c)

49 543

(d)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials 453

The number of values of k for which the equation x 3 - 3x + k = 0 has two distinct roots lying in the interval (0, 1) is three (b) two (c) infinitely many (d) zero  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 454

Consider the function f(x) = xcosx - sinx. Then identify the statement which is correct. f is neither odd nor even. f is monotonic decreasing at x = 0 f has a maxima at x = π f has a minima at x = - π  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials 455

Let f(x) = ax 3 + bx 2 + cx + 1 has exterma at x = α, β such that αβ < 0andf(α)f(β) < 0 .

Then the equation f(x) = 0 has three equal real roots one negative root if f(α)〈0andf(β)〉0 one positive root if f(α)〈0andf(β)〉0 none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

456

A factory D is to be connected by a road with a straight railway line on which a town A is situated. The distance DB of the factory to the railway line is 5√ 3km. Length AB of the railway line is 20km. Freight charges on the road are twice the charges on the railway. The point `P(A P  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

457

OF

DERIVATIVES_Solid

The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm π π π and semi-vertical angle 30 0 is (a)4000 (b) 400 cm 3 4000 (d) none of these 3 3 3 √ √3cm  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

458

OF

DERIVATIVES_Solid

A rectangle of the greatest area is inscribed in a trapezium ABCD , one of whose nonparallel sides AB is perpendicular to the base, so that one of the rectangles die lies on the larger base of the trapezium. The base of trapezium are 6cm and 10 cm and AB is 8 cm long. Then the maximum area of the rectangle is 24cm 2 (b) 48cm 2 36cm 2 (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

459

OF

DERIVATIVES_Solid

A bell tent consists of a conical portion above a cylindrical portion near the ground.

For a given volume and a circular base of a given radius, the amount of the canvas cos - 12 sin - 1 2 used is a minimum when the semi-vertical angle of the cone is (b)

3 3 cos - 1 1 (d) none of these 3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

OF

DERIVATIVES_Solid

A rectangle is inscribed in an equilateral triangle of side length 2a units. The maximum 460

area of this rectangle can be √ 3a2 (b)

√ 3a 4

2

a2 (d)

√ 3a

2

2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

461

Tangents are drawn to x 2 + y 2 = 16 from the point P(0, h). These tangents meet the x - aξs at AandB . If the area of triangle PAB is minimum, then h = 12√2 (b) h = 6√ 2 h = 8√2 (d) h = 4√ 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

462

OF

DERIVATIVES_Solid

The largest area of the trapezium inscribed in a semi-circle or radius R, if the lower 3√ 3 √ 3 2 3√ 3 2 2 base is on the diameter, is R (b) R R (d) R 2 4 2 8  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

463

OF

DERIVATIVES_Solid

π In a ABC, ∠B = 90 0 andb + a = 4. The area of the triangle is maximum when ∠C is (b) 4 π π (d) none of these 6 3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

OF

DERIVATIVES_Solid

464

The three sides of a trapezium are equal, each being 8cm. The area of the trapezium, where it is maximum, is (a)24√ 3 cm 2 (b) 48√ 3cm 2 (c)72√3cm 2 (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

465

The fuel charges for running a train are proportional to the square of the speed generated in km/h, and the cost is Rs. 48 at 16 km/h. If the fixed charges amount to Rs. 300/h, the most economical speed is 60 km/h (b) 40 km/h 48 km/h (d) 36 km/h  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

466

OF

DERIVATIVES_Solid

A cylindrical gas container is closed at the top and open at the bottom. If the iron plate of the top is 5/4 times as thick as the plate forming the cylindrical sides, the ratio of the radius to the height of the cylinder using minimum material for the same capacity is 3:4 (b) 5:6 (c) 4:5 (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 467

The least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is 4√ 3r (b) 2√3r 6√3r (d) 8√3r  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry 468

OF

DERIVATIVES_Solid

A given right cone has volume p, and the largest right circular cylinder that can be inscribed in the cone has volume q. Then p : q is 9:4 (b) 8:3 (c) 7:2 (d) none of these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

469

A wire of length a is cut into two parts which are bent, respectively, in the form of a a2 square and a circle. The least value of the sum of the areas so formed is (b) π+4 a a a2

(d) π + 4 4(π + 4) 4(π + 4)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 470

A box, constructed from a rectangular metal sheet, is 21 cm by 16cm by cutting equal squares of sides x from the corners of the sheet and then turning up the projected portions. The value of x os that volume of the box is maximum is (a)1 (b) 2 (c) 3 (d)4  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum The vertices of a triangle are (0,0), (x, cosx), and 471 maximum area for such a triangle in sq. units is (a)

3√ 3 32

(

sin3 x, 0

(b)

), where0 < x < 2 the

( ) √3 32

π

(c)

4 32

(d)

6√ 3 32

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 472

The maximum area of the rectangle whose sides pass through the vertices of a given 1 1 rectangle of sides aandb is (a)2(ab) (b) (a + b) 2 (c) a2 + b2 (d) noneofthese 2 2

(

)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

473

OF

DERIVATIVES_Solid

The base of prism is equilateral triangle. The distance from the centre of one base to one of the vertices of the other base is l . Then altitude of the prism for which the l l l l volume is greatest is (b) (c) (d) 2 3 4 √3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 474

`L e tf(x)={x^2+3x ,-1lt=x<0-sinx ,0lt=x  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 475

Leg f(x) = x 4 - 4x 3 + 6x 2 - 4x + 1. Then, f increase on [1, ∞] f decreases on [1, ∞] f has a minimum at x = 1 f has neither maximum nor minimum  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 476

OF

Let f(x) = 2x - sinxandg(x) = x3 . Then A. range of gof is R B. gof is one-one C. both f and g are one-one D. both f and g are onto  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

477

If f(x) = 2x + cot - 1 x + log

(√

)

OF

1 + x2 - x then f(x) (a)increase in (0, ∞) (b)decrease in

[0, ∞] (c)neither increases nor decreases in [0, ∞] (d)increase in ( - ∞, ∞)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

|

[ ]

|

Let f(x) = x 2 - 3x - 4 , - 1 ≤ x ≤ 4 Then f(x) is monotonically increasing in - 1, 478 monotonically decreasing in

( ) 3 2

, 4 the maximum value of f(x)is

25 4

3 2

f(x)

the minimum value

of f(x) is 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum x sint If f(x) = ∫ 0 dt, x > 0, then (a)f(x) has a local maxima at x = nπ n = 2k, k ∈ I + (b)f(x) t

(

479

(

)

)

has a local minima at x = nπ n = 2k, k ∈ I + (c)f(x) has neither maxima nor minima at

(

)

(

x = nπ n ∈ I + (d)(f)x has local maxima at x = nπ n = 2k, 1, k ∈ I +

)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous

480

OF

The values of parameter a for which the point of minimum of the function x2 + x + 2 2 3 f(x) = 1 + a x - x satisfies the inequality < 0are 2√ 3, 3√ 3 (b) 2 x + 5x + 6

(

)(

)

(

- 3√ 3, - 2√ 3 - 2√3, 3√ 3 (d) - 2√2, 2√3  Watch Free Video Solution on Doubtnut

)

)

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 481

Let f(x) = ax 2 - b|x|, where aandb are constants. Then at x = 0, f(x) has a maxima whenever a > 0, b > 0 a maxima whenever a > 0, b < 0 minima whenever a > 0, b < 0 neither a maxima nor a minima whenever a > 0, b < 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

482

The function y =

OF

2x - 1

, (x ≠ 2) (a)is its own inverse (b)decrease at all values of x in the x-2 domain (c)has a graph entirely above the x - axis (d)is unbounded.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous 483

OF

Let g ′ (x) > 0andf ′ (x) < 0 ∀x ∈ R. Then (f(x + 1)) > g(f(x - 1)) f(g(x - 1)) > f(g(x + 1)) `g(f(x+1))  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials

484

If f(x) = x 3 - x 2 + 100x + 2002, then f(1000) > f(1001) f

( ) ( ) 1

2000

>f

1

2001

f(x - 1) > f(x - 2)

f(2x - 3) > f(2x)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

485

If f ′ x = g(x)(x - a) 2 , whereg(a) ≠ 0, andg is continuous at x = a, then f is increasing in the neighbourhood of a if g(a) > 0 f is increasing in the neighbourhood of a if g(a) < 0 f is decreasing in the neighbourhood of a if g(a) > 0 f is decreasing in the neighbourhood of a if g(a) < 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum The value of a for which the function f(x) = (4a - 3)(x + log5) + 2(a - 7) 486

(

not possess critical points is (a) - ∞, -

4 3

)

cotx sin 2x 2

2

does

(b) ( - ∞, - 1) (c)[1, ∞) (d) (2, ∞)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

487

Let f(x) = (x - 1) 4(x - 2) n, n ∈ N. Then f(x) has a maximum at x = 1 if n is odd a maximum at x = 1 if n is even a minimum at x = 1 if n is even a minima at x = 2 if n is even  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous

488

OF

Let f(x) = sinx + ax + b. Then which of the following is/are true? f(x) = 0 has only one real root which is positive if a > 1, b < 0. f(x) = 0 has only one real root which is negative if a > 1, b < 0. f(x) = 0 has only one real root which is negative if a > 1, b > 0. noneofthese  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Integration As Reverse Process Of Differentiation

489

sin(x + a)

has no maxima or minima if (a) b - a = nπ, n ∈ I (b) sin(x + b) b - a = (2n + 1)π, n ∈ I (c) b - a = 2nπ, n ∈ I (d) none of these The function

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous

OF

( ( (( )))

490

If composite function f 1 f 2 f 3 f n(x) n timesis an increasing function and if r of f i′ s are decreasing function while rest are increasing, then the maximum value of r(n - r) is n2 - 1 n2 n2 - 1 , when n is an even number , when n is an odd number , when n is an 4 4 4 n2 odd number , when n is an even number 4  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

491

{

(x - 1)(6x - 1)

1

1

,

if x ≠

0,

if x =

Then at x =

1

, which of the following 2x - 1 2 7 2 is/are not true? f has a local maxima f has a local minima f has an inflection point. f has a removable discontinuity.

Letf(x) =

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous 492

OF

(

) (

)

Let f(x) be an increasing function defined on (0, ∞) . If f 2a 2 + a + 1 > f 3a 2 - 4a + 1 , then the possible integers in the range of a is/are 1 (b) 2 (c) 3 (d) 4  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

493

(

If f(x) = sin 2x - 1

)

n

, then x =

π

is a point of local maximum, if n is odd local minimum, 2 if n is odd local maximum, if n is even local minimum, if n is even  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

494

OF

For the cubic function f(x) = 2x 3 + 9x 2 + 12x + 1, which one of the following statement/statements hold good? f(x) is non-monotonic. f(x) increases in → ( - ∞, - 2) ∪ ( - 1, ∞) and decreases in ( - 2, - 1) f : RR is bijective. Inflection point occurs 3 at x = - .

2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 495

Let f(x) = a 5 x 5 + a 4x 4 + a 3x 3 + a 2 x2 + a1 x, where a i′ s are real and f(x) = 0 has a positive root α 0. Then f ′ (x) = 0 has a positive root α 1 such that `0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

496

OF

If f(x)andg(x) are two positive and increasing functions, then which of the following is not always true? [f(x)] g ( x ) is always increasing [f(x)] g ( x ) is decreasing, when f(x) < 1 [f(x)] g ( x ) is increasing, then f(x) > 1. If f(x) > 1, then[f(x)] g ( x ) is increasing.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum

497

An extremum of the function f(x) =

2-x π

cosπ(x + 3) +

x = 1 (b) x = 2 x = 3 (d) x = π  Watch Free Video Solution on Doubtnut

1 π

2

sinπ(x + 3), 0x < 4, occurs at

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

498

For the function f(x) =

x4

(12(log)ex - 7 ), the point (1,7) is the point of inflection. x = e

1 3

is the point of minima the graph is concave downwards in (0,1) the graph is concave upwards in (1, ∞)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

499

(

)

Let f(x) = log 2x - x 2 +

sin(πx)

. Then which of the following is/are true? Graph of f is

2 symmetrical about the line x = 1 Maximum value of fis1. Absolute minimum value of f does not exist. none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 2

500

Which of the following hold(s) good for the function f(x) = 2x - 3x 3 ? (a)f(x) has two points of extremum. (b)f(x) is convave upward ∀x ∈ R. (c)f(x) is non-differentiable function. (d)f(x) is continuous function.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous For the function f(x) = 501

ex x

OF

, which of the following hold good? f is monotonic in its

1+e entire domain. Maximum of f is not attained even though f is bounded f has a point of inflection. f has one asymptote.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection

502

Which of the following is true about point of extremum x = a of function y = f(x) ? At x = a, function y = f(x) may be discontinuous. At x = a, function y = f(x) may be continuous but non-differentiable. At x = a, functiony = f(x) may have point of inflection. none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum Which of the following function has point of extremum at x = 0 ? f(x) = e - | x | f(x) = sin|x| 503

f(x) =

{x

2

+ 4x + 3, x < 0 - x, x ≥ 0 f(x) = {|x|, x < 0{x}, 0 ≤ x < 1 (where {x} represents

fractional part function).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 6

504

Which of the following function/functions has/have point of inflection? f(x) = x 7 (b) f(x) = x 6 f(x) = cosx + 2x (d) f(x) = x|x|  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection

505

λ The function f(x) = x 2 + has a (a)minimum at x = 2 if λ = 16 (b)maximum at x x = 2 if λ = 16 (c)maximum for no real value of λ (d)point of inflection at x = 1 if λ = - 1  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection 1

506

The function f(x) = x 3 (x - 1) has two inflection points has one point of extremum is nondifferentiable has range

[

8

- 3x2 - 3 ,

∞

)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum

507

( )

Let f be the function f(x) = cosx - 1 -

x2 2

. Then f(x) is an increasing function in (0, ∞)

f(x) is a decreasing function in ( - ∞, ∞) f(x) is an increasing function in ( - ∞, ∞) f(x) is a decreasing function in ( - ∞, 0)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

508

Statement 1: Both sinxandcosx are decreasing functions in

OF

( ) π

, π Statement 2: If a 2 differentiable function decreases in an interval (a, b), then its derivative also decreases in (a, b) .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 509

Statement 1: `alpha^(beta)>beta^(alpha),F O R2. 91 edot`  Watch Free Video Solution on Doubtnut

OF

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 510

Statement 1: f(x) = |x - 1| + |x - 2| + |x - 3| has point of minima at x = 3. Statement 2: f(x) is non-differentiable at x = 3.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

511

Statement 1: The function f(x) = x In x is increasing in

OF

( ) 1

, ∞ Statement 2: If both e f(x)andg(x) are increasing in (a, b), thenf(x)g(x) must be increasing in (a,b).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

OF

→

512

Let f: RR be differentiable and strictly increasing function throughout its domain.

Statement 1: If |f(x)| is also strictly increasing function, then f(x) = 0 has no real roots. → → → Statement 2: When x∞ or - ∞, f(x) 0 , but cannot be equal to zero.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

OF

2

513

Statement 1: Let f(x) = 5 - 4(x - 2) 3 . Then at x = 2, the function f(x) attains neither the least value nor the greatest value. Statement 2: At x = 2, then first derivative does not exist.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 514

OF

Statement 1: f(x) = x + cosx is increasing ∀ x ∈ R. Statement 2: If f(x) is increasing, then f ′ (x) may vanish at some finite number of points.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous

OF

515

sin - 1x

Statement 1: Both f(x) = 2cosx + 3sinxandg(x) =

√13

-

tan - 13 2

are increasing in (0,π/2)

Statement 2: If f(x) is increasing, then its inverse is also increasing.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials

516

Statement 1: f(x) =

x3

ax 2

+ + x+5 has positive point of maxima for a < - 2.

3 2 Statement 2: x 2 + ax + 1 = 0 has both roots positive for a < 2.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 517

Statement 1: For all a, b ∈ R, the function f(x) = 3x 4 - 4x 3 + 6x 2 + ax + b has exactly one extremum. Statement 2: If a cubic function is monotonic, then its graph cuts the xaxis only one.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

518

[

Statement 1: The value of (lim) n ∞ →

sinxtanx 2

]

x integer function. Statement 2: For `(0,pi/2),sinx  Watch Free Video Solution on Doubtnut

is 1, where [.] denotes the greatest

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

[ ] [ ]

Statement 1: Let f(x) = sin(cosx) ∈ 0, 519

π 2

2: cosx is a decreasing function ∀x ∈ 0,

. Then f(x) is decreasing in

[ ] 0,

π 2

Statement

π 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials

520

(

)

Let f(x) = x 3 - 6x 2 + 12x - 8 e x. Statement 1: f(x) is neither maximum nor minimum at x = 2. Statement 2: If a function x = 2 is a point of inflection, then it is not a point of extremum.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 521

OF

Statement 1: The function f(x) = x 4 - 8x3 + 22x2 - 24x + 21 is decreasing for every x ∈ ( - ∞, 1) ∪ (2, 3) Statement 2: f(x) is increasing for x ∈ (1, 2) ∪ (3, ∞) and has no point of inflection.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

522

(

Statement 1: If f(0) = 0, f ′ (x) = ln x +

√

OF

)

1 + x 2 , then f(x) is positive for all x ∈ R0 .

Statement 2: f(x) is increasing for x > 0 and decreasing for x < 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION

OF

DERIVATIVES_Miscellaneous

523

The function f(x) =

√ (ax 3 + bx 2 + cx + a ) ha sits non-zero local minimum and local

maximum values at x - 2 and x = 2, respectively. It 'a is a root of x 2 - x - 6 = 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous

OF

If α is an integer satisfying |α| ≤ 4 - |[x]|, where x is a real number for which 2xtan - 1x is 524

(

)

greater than or equal to ln 1 + x 2 , then the number of maximum possible values of α (where [.] represents the greatest integer function) is_____  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

525

{

(

OF

DERIVATIVES_Solid

)

1 x 3 + x 2 + 3x + sinx ∣ 3 + s ∈ , , x ≠ 0. 0x = 0 then the number of point x where f(x) attains its minimum value is_____ Letf(x) =

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 526

Leg f(x) be a cubic polynomial which has local maximum at x = - 1andf(x) has a local minimum at x = 1. If f( - 1) = 10andf(3) = - 22, then one fourth of the distance between its two horizontal tangents is ____________  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials Consider P(x) to be a polynomial of degree 5 having extremum at x = - 1, 1, and 527

(lim) x0 →

( ) p(x) 3

x function)___

- 2 = 4 . Then the value of [P(1)] is (where [.] represents greatest integer

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 528

If m is the minimum value of f(x, y) = x 2 - 4x + y 2 + 6y when xandy are subjected to the restrictions 0 ≤ x ≤ 1and0 ≤ y ≤ 1, then the value of |m| is________  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum For a cubic function y = f(x), f x = 4x at each point (x, y) on it and it crosses the x - aξs at 529

( - 2, 0) at an angle of 45 0 with positive direction of the x-axis. Then the value of

| | f(1) 5

is_______  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

530

Number of integral values of b for which the equation solutions is____

x3 3

- x = b has three distinct

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials 531

Letf(x) =

{x + 2, x < - 1; x , - 1 ≤ x < 1 and (x - 2) , x ≥ 1 Then number of times f (x) 2

changes its sign in ( - ∞, ∞) is___  Watch Free Video Solution on Doubtnut

2

′

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection

532

The number of nonzero integral values of a for which the function 3x 2 4 3 f(x) = x + ax + + 1 is concave upward along the entire real line is___________ 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Point Of Inflection

533

Let f(x) =

{

3

x 5,

if x ≤ 1 and (x - 2) 3,

if x > 1 Then the number of critical points on

the graph of the function is___  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

534

1 A right triangle is drawn in a semicircle of radius with one of its legs along the 2 diameter. If the maximum area of the triangle is M , then the value of 32√3Mc is______  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

535

A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines y = 0, y = 3x, andy = 30 - 2x. If M is the largest 2M area of such a rectangle, then the value of is_______ 27  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 536

OF

(

)

The least integral value of x where f(x) = (log) 1 x 2 - 2x - 3 is monotonically 2

decreasing is________  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 537

9 The least area of a circle circumscribing any right triangle of area is ___________ π  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

538

Letf(x) =

{|

|

x 2 - 3x + a, 0 ≤ x <

3

- 2x + 3, x ≥

2 then greatest value of |4a| is _________

3 2

If f(x) has a local maxima at x =

3 2

,

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 539

Prove that the minimum value of

(a + x)(b + x) (c + x)

a, b > c, x ≻ c is

(√a - c + √b - c ) 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application

Of Extremum 540

Let xandy be two real variable such that x > 0andxy = 1 . Find the minimum value of x + y .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 541

1

Use the function f(x) = x x , x > 0, to determine the bigger of the two numbers e π andπ e.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 542

Find the shortest distance of the point (0, c) from the parabola y = x 2 , where0 ≤ c ≤ 5.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 543

If ax 2 +

b x

≥ c for all positive x where a > 0 and b > 0, show that 27ab2 ≥ 4c 3 .

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 544

(

Show that 1+x ln x +

OF

√x 2 + 1 ) ≥ √1 + x2 for all x ≥ 0.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

545

A swimmer S is in the sea at a distance dkm from the closest point A on a straight shore. The house of the swimmer is on the shore at distance Lkm from A. He can swin at speed oh ukm/h. walk at speed at vkm/h. At what point on shore should he land so that hhe reaches his house in the shortest time.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

546

Find the coordinates of the point on the curve y = curve has the greatest slope.

x 2

1+x

where the tangent to the

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 547

OF

Let `f(x)=sin^3x+lambdasin^2x ,-pi/2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

(

) ( ) (

)

Let A p2 , - p , B q 2, q , C r 2, - r be the vertices of triangle ABC. A parallelogram 548

AFDE is drawn with D,E, and F on the line segments BC, CA and AB, respectively.

Using calculus, show that the maximum area of such a parallelogram is 1 (p + q)(q + r)(p - r).

2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 549

Find the point on the curve `4x^2+a^2y^2=4a^2,4  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum Investigate 550

for

the

maxima

and

minima

of

the

function

x

[

]

f(x) = ∫ 1 2(t - 1)(t - 2) 3 + 3(t - 1) 2 (t - 2) 2 dt  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 551

Find the maxima and minima of the function y = x(x - 1) 2, 0 ≤ x ≤ 2.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 552

OF

Show that `2sinx+tanxgeq3x ,w h e r e0lt=x  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 553

A point P is given on the circumference of a circle of radius r . Chords QR are parallel to the tangent at P. Determine the maximum possible area of triangle PQR.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

554

A window of perimeter P (including the base of the arch) is in the form of a rectangle surrounded by a semi-circle. The semi-circular portion is fitted with the colored glass while the rectangular part is fitted with the clear glass that transmits three times as much light per square meter as the colored glass does. What is the ratio for the sides

of the rectangle so that the window transmits the maximum light?  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials

555

A cubic function f(x) vanishes at x = - 2 and has relative minimum/maximum at 1 14 1 . Find the cubic function f(x). x = - 1andx = if ∫ - 1 f(x)dx = 3 3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

556

Letf(x) =

{ (( - x3 +

b 3 - b2 + b - 1

)

Find all the possible real

b 2 + 3b + 2 2x - 3, 1 ≤ x ≤ 3, 0 ≤ x < 1

values of b such that f(x) has the smallest value at x = 1.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 557

The circle x 2 + y 2 = 1 cuts the x-axis at PandQ . Another circle with center at Q and variable radius intersects the first circle at R above the x-axis and the line segment PQ at S. Find the maximum area of triangle QSR.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 558

Let (h, k) be a fixed point, where h > 0, k > 0. A straight line passing through this point cuts the positive direction of the coordinate axes at the point PandQ . Find the minimum area of triangle OPQ, O being the origin.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 559

Determine the point of maxima and minima 1 f(x) = (log) ex - bx + x 2 , x > 0, where b ≥ 0 is constant.

8

of

the

function

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 560

Letf(x) =

{xe

ax,

OF

x ≤ 0x + ax 2 - x 3 , x > 0 where a is a positive constant. Find the interval

in which f ′ (x) is increasing.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous

561

OF

Suppose f(x) is a function satisfying the following conditions: f(0) = 2, f(1) = 1 f has a 5 minimum value at x= For all 2 x, f ′ (x) = |2ax2ax - 12ax + b + 1 + 1 - 12(ax + b)2ax + 2b + 12ax + b| where a, b are some constants. Determine the constants a, b , and the function f(x)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials 562

Let - 1 ≤ p ≤ 1 . Show that the equation 4x 3 - 3x - p = 0 has a unique root in the interval [1/2,1] and identify it.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 563

Using the relation `2(1-cosx)  Watch Free Video Solution on Doubtnut

OF

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 564

If P(1) = 0and

dP(x) dx

> P(x), for all x ≥ 1. Prove that P(x) > 0 for all x > 1.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 565

If P(x) is a polynomial of degree 3 satisfying p( - 1) = 10, p(1) = - 6andp(x) has maxima at x = - 1andp ′ (x) has minima at x = 1, find the distance between the local maxima and local minima of the curve.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum π

566

The larger of cos(lnθ) and ln(cosθ) if e - 2

<θ<

π 2

is

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 567

The function y = 2x 2 - ln|x| is monotonically increasing for values of x( ≠ 0) satisfying the inequalities____ and monotonically decreasing for values of x satisfying the inequalities_____.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 568

The set of values for which (log) e(1 + x) ≤ x is equal to_______

OF

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Concept Of Global Maximum/Minimum 569

The maximum value of the function f(x) = 2x 3 - 15x 2 + 36x - 48 on the set A=

{x |x |20 ≤ 9x} is______.

2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 570

OF

3

If f(x) = x 2 (3x - 10), x ≥ 0, then f(x) is increasing in ____.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 571

If x - r is a factor of the polynomial f(x) = a 0x n + a 1x n - 1 + .... . + an repeated m times, (1 < x ≤ n), then r is a root of f ′ (x) = 0repeated m times.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 572

(

)

For 0 < a < x, the minimum value of function (log) a = x + log xa = a is 2.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 573

AB is a diameter of a circle and C is any point on the circumference of the circle. Then the area of ABC is maximum when it is isosceles the area of ABC is minimum when it is isosceles the perimeter of ABC is minimum when it is isosceles none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 574

OF

If f(x) = alog|x| + bx 2 + x has its extremum values at x = - 1andx = 2, then  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum The function f(x) = 575

( ) 0,

π

e

ln(π + x) ln(e + x)

, decreasing in

is increasing in (0, ∞) decreasing in (0, ∞) increasing in

( ) π

e

( )

, ∞ decreasing in 0,

π

e

, increasing in

( ) π

e

,∞

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

OF

In the interval [0, 1], the function x 25(1 - x) 75 takes its maximum value at the point 0 (b) 576

1 4

1

1

(c) (d) 2 3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 577

If f(x) =

x sinx

andg(x) =

x tanx

OF

, where0 < x ≤ 1, then in this interval

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 578

OF

The function f(x) = sin 4x + cos 4x increasing if `0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

OF

Consider the following statements in S and R S: Both sinxandcosx are decrerasing function in the interval

( ) π 2

, π R: If a differentiable function decreases in an interval

)

579

(a, b), then its derivative also decrease in a, b . Which of the following it true? Both S and R are wrong. Both S and R are correct, but R is not the correct explanation of S. S is correct and R is the correct explanation for S. S is correct and R is wrong.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 580

Let f(x) = ∫ e x(x - 1)(x - 2)dx. Then f decreases in the interval (a)( - ∞, - 2) (b) - 2, - 1) (c) (1, 2) (d) (2, + ∞)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 581

OF

Letf(x) = {|x|, f or 0 < |x| ≤ 2 : 1, forx = 0 Then at x = 0, f has (a)a local maximum (b) no local maximum (c)a local minimum (d) no extremum  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 582

For all x ∈ (0, 1) e x < 1 + x (b) `(log)_e(1+x) x(d)(log)_e x > x`  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

583

[ ]

1 If f(x) = xe x ( x - 1 ) , then f(x) is increasing on - , 1 decreasing on R increasing on R 2

[ ]

(d) decreasing on -

1 2

,1

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions

(

OF

)

Let f(x) = 1 + b 2 x 2 + 2bx + 1 and let m(b) the minimum value of f(x) . As b varies, the 584

( ][ ]

range of m(b) is [0, 1] (b) 0,

1 2

1 2

, 1 (d) (0, 1]

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum The length of the longest interval in which the function 3sinx - 4sin 3x is increasing is 585

π 3π (b) (c) (d) π 2 2  Watch Free Video Solution on Doubtnut

π 3

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Nature Of Roots Of Cubic Polynomials y2

(

)

( )] π

+ = 1 at 3√3cosθ, sinθ [where θ ∈ 0, Then 27 2 the value of θ such that sum of intercepts on axes made by this tangent is minimum is π π π π (b) (c) (d) 3 6 8 4 Tangent is drawn to ellipse

586

x2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 587

If `f(x)=x^3+b x^2+c x+da n d0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 588

OF

OF

The function defined by f(x) = (x + 2)e - x is (a)decreasing for all x (b)decreasing in ( - ∞, - 1) and increasing in ( - 1, ∞) (c)increasing for all x (d)decreasing in ( - 1, ∞) and increasing in ( - ∞, - 1)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 589

OF

→ The function f : [0, 3]1, 29, defined by f(x) = 2x3 - 15x 2 + 36x + 1, is one-one and onto onto but not one-one one-one but not onto neither one-one nor onto  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

590

The number of points in ( - ∞, ∞), for which x 2 - xsinx - cosx = 0, is 6 (b) 4 (c) 2 (d) 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 591

Let P(x) = a 0 + a1 x 2 + a 2x 4 + + a nx 2n be a polynomial in a real variable x with `0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Miscellaneous

OF

( ) ( )( )

The smallest positive root of the equation tanx - x = 0 lies in 0, 592 (d)

π 2

(b)

π 2

, π π,

3π 2

( ) 3π 2

, 2π

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 593

OF

Let fandg be increasing and decreasing functions, respectively, from [0, ∞] → [0, ∞].

Let h(x) = f(g(x)). If h(0) = 0, then h(x) - h(1) is always zero (b) always negative always positive (d) strictly increasing none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 594

`Iff(x)={3x^2+12 x-1,-1lt=xlt=2 37-x ,2  Watch Free Video Solution on Doubtnut

OF

CENGAGE_MATHS_CALCULUS_APPLICATION DERIVATIVES_Monotonocity : Introductions 595

OF

Let h(x) = f(x) - (f(x)) 2 + (f(x)) 3 for every real number x. Then h is increasing whenever f is increasing h is increasing whenever f is decreasing h is decreasing whenever f is decreasing nothing can be said in general  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum x2 - 1

596

If f(x) = 2 . For every real number x, then the minimum value of f . does not exist x +1 because f is unbounded is not attained even through f is bounded is equal to 1 is equal to - 1  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 597

The number of values of x where the function f(x) = cosx + cos

(√2x ) attains its

maximum is 0 (b) 1 (c) 2 (d) infinite  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 598

(

)

x The function f(x) = ∫ - 1 t e t - 1 (t - 1)(t - 2) 3(t - 3) 5dt has a local minimum at x = 0 (b) 1 (c) 2 (d) 3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

599

f(x) is cubic polynomial with f(x) = 18andf(1) = - 1 . Also f(x) has local maxima at x = - 1andf ′ (x) has local minima at x = 0 , then the distance between ( - 1, 2)and(af(a)),

[

)

where x = a is the point of local minima is 2√5 f(x) is increasing for x ∈ 1, 2√5 f(x) has local minima at x = 1 the value of f(0) = 15  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 600

`f(x)={c^x ,0lt=x1 2-c^(x-1),1  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum

601

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into anopen rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the length of the sides of the rectangular sheet are 24 (b) 32 (c) 45 (d) 60  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

602

The function f(x) = 2|x| + |x + 2| = ||x|2| - 2|x|∣ has a local minimum or a local maximum at 2 2 x = - 2 (b) - (c) 2 (d) 3 3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 603

For every pair of continuous functions f, g : [0, 1] → R such that max {f(x) : x ∈ [0, 1]} = max {g(x) : x ∈ [0, 1]} then which are the correct statements  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Application Of Extremum 604

Late a ∈ R and let f : R→ be given by f(x) = x 5 - 5x + a, then f(x) has three real roots if a > 4 f(x) has only one real roots if a > 4 f(x) has three real roots if a < - 4 f(x) has

three real roots if `-4  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum

605

Let f be a function defined on R (the set of all real numbers) such that f ′ (x) = 2010(x - 2009)(x - 2010) 2 (x - 2011) 3 (x - 2012) 4 , for all x ∈ R. If g is a function defined on R with values in the interval (0, ∞) such that f(x) = ln(g(x)), for all x ∈ R, then the number of point is R at which g has a local maximum is ___  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 606

|

|

→ Let IR I R be defined as f(x) = |x| + x 2 - 1 . The total number of points at which f attains

either a local maximum or a local minimum is_______  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION OF DERIVATIVES_Tests For Local Maximum/Minimum 607

Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and a local minimum at x = 3. If p(1) = 6andp(3) = 2, then p ′ (0) is_____  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_CALCULUS_APPLICATION Geometry

608

OF

DERIVATIVES_Solid

A cylindrical container is to be made from certain solid material with the following constraints: It has fixed inner volume of Vmm 3 , has a 2mm thick solid wall and is open at the top. The bottom of the container is solid circular disc of thickness 2mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then

the value of

V 250π

is

 Watch Free Video Solution on Doubtnut

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