f4

Page 31

Solutions, Section 4.5

31

√

x 3 − 3x2 . f (x) = 0 for x = −1, x = 0 and x = 1, however, x = 0 , f (x) = √ +3 2 x(x2 + 3)2 and x = −1 are outside the interval, thus f has a maximum of 1/4 at x = 1 (ďŹ rst derivative test). There is no minimum.

4.5.11 f (x) =

x2

1 − x2 x (x) = . f (x) = 0 when x = −1 and x = 1, however, x = −1 is ; f x2 + 1 (x2 + 1)2 outside the interval. f (0) = 0; f (1) = 1/2; and f (2) = 2/5, so f has a maximum of 1/2 at x = 1 and a minimum of 0 at x = 0.

4.5.12 (x) =

1 x so there are no critical points. f (0) = 0, thus, for x in [0, +∞); f (x) = 1+x (1 + x)2 f has a minimum of 0 at x = 0. There is no maximum.

4.5.13 f (x) =

2x − 1 1 . f (x) = 0 when x = 1/2; f (x) does not exist at x = 0 , f (x) = x − x2 (x − x2 )2 or x = 1, however, both of these values are outside the interval. f has a minimum of +4 at x = 1/2, there is no maximum.

4.5.14 f (x) =

4.5.15 f (x) =

x2 ,

x<0

3

x≼0

x ,

f (x) =

2x,

x<0

2

x>0

3x ,

f (x) = 0 when x = 0 which corresponds to a minimum value (ďŹ rst derivative test), there is no maximum.   x < −1 x < −1    x − 1,  −1, 2 1 − x , −1 ≤ x ≤ 1 −2x, −1 < x < 1 4.5.16 f (x) = f (x) =     x − 1, x>1 1, x>1 f (x) = 0 when x = 0, f (x) does not exist at x = −1 or x = 1. f (−2) = 1; f (−1) = 0; f (0) = 1; f (1) = 0; f (2) = 1, thus, f has a maximum of 1 at x = −2, x = 0, and x = 2; f has a minimum of 0 at x = −1 and x = 1. 1 − x2 , x < 0 −2x, x < 0 4.5.17 f (x) = f (x) = 3 3x2 , x > 0 x − 1, x ≼ 0 f (x) = 0 when x = 0. f (−2) = −3, f (0) = −1, f (1) = 0, thus, f has a maximum of 0 at x = 1 and a minimum of −3 at x = −2. −2, x < 3/2 3 − 2x, x ≤ 3/2 4.5.18 f (x) = 3 − 2x = f (x) = 2, x > 3/2 −3 + 2x, x > 3/2 3 = 0, f (2) = 1, thus, f has a maximum of f (x) does not exist at x = 3/2. f (−2) = 7, f 2 7 at x = −2 and a minimum of 0 at x = 3/2. Ď€ 2 Absolute minimum of 0 at 0

4.5.19 Absolute maximum of 4 at

4.5.20 Absolute maximum of 16 at 2 Absolute minimum of 0 at 0

4.5.21 Absolute maximum of 8.43 at 1.78 Absolute minimum of 0 at 0

4.5.22 Absolute maximum of 3.56 at 0.86 Absolute minimum of 0 at 0

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