If f is a continuous function on the closed interval [a, b], which of the following statements are NOT necessarily true?

I. f has a minimum on [a, b]
II. f has a maximum on [a, b]
III f ‘(c) = 0 for a

Question

If f is a continuous function on the closed interval [a, b], which of the following statements are NOT necessarily true? 

I. f has a minimum on [a, b] 
II. f has a maximum on [a, b] 
III f ‘(c) = 0 for a < c < b

anonymous · Answer

why on earth would the derivative of a continuous function be zero?

anonymous · Answer

a continuous function may not even have a derivative

aravindg · Answer

Clearly III f ‘(c) = 0 for a < c < b 
Only holds for Rolle's theorem where f(a)=f(b)

aravindg · Answer

So statement 3 is not always necessary.