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

Answer 1

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

Answer 2

a continuous function may not even have a derivative

Answer 3

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

Answer 4

So statement 3 is not always necessary.