Drawing Curves
E4) a) Y q y ; = 2 s i n x c o s x = s i n 2 ~ = 0 i f x = d 2 [O,IC] ~ b) Yes,f (x)=2x=0 ifxaO E [-2,2] c) No. f (x) = 3x2+ 1 # 0 for x E [0, 11. Rolle's t h e ~ r ~ d onot e shold as 40)#4l)
E 5) c = 3 0 . Yes.
E8) Suppose f(x) is not (1 - 1) on [boo [ *p,q[xo,=[,suchthatp*qandf@)=f(q)
P*9 = x,,as x,, 2
is the unique point with f (x,,)~ 0 Therefore either p c x,,or q c x, since p and q both cannot be equal to x,,. This is a contradiction as we have taken, p,q E [x, a [. E9) Suppose p,q e I s.t. p # q and f@)= f(q) If p < q we have [p, ql c I, f is differentiableon [p, ql and f@)= f(q). Thus f satisfiesthe conditions of Rolle's theorem on [p, ql ~f(x,,)=Oforsomex,,e[p,qlcl. But this is a contradiction Therefore f is one-one
E l l ) a) qO)=O=f12) f (x)=~cosmaf(1/2)=0.