The function f is continuous and non-linear for -3 is equal and less than x is equal and less than 7 and f(-3)=5 and f(7)=-5.

Question

If there is no value of c on -3<x<7 for which f'(c)=-1, which of the following statements must be true?

a) for come c on -3<c<7, f'(c)<-1

b) For some c on -3<c<7, f'(c)>-1 c

) for some c on -3<c<7, f'(c)=0

d) For -3<c<7, f'(c) exists

e) for some c on -3<c<7, f'(c) does not exist

 

Answer 1

By assumption, f is continuous on [-3, 7].  If f is also differentiable on (-3, 7), then, by the Mean Value Theorem, there would exist at least one number c where -3 < c < 7 and for which

f'(c) = [f(7) - f(-3)]/[7 - (-3)] = -1.  Since no such number exists (by assumption), f must not be differentiable on the interval (-3, 7).  

 

ANSWER:  (e)