7. Let f be a function that is continuous on the closed interval [0,1] and differentiable on the open interval (0,1). If f(0) = f(1), then which of the following statements must be true?

 

A.

f has a minimum at some x0 such that 0 < x0 < 1.

B.

f has a maximum at some x0 such that 0 < x0 < 1.

C.

f’ has a minimum at some x0 such that 0 < x0 < 1.

D.

f’(x0) = 0 at every x0 such that 0 < x0 < 1.

E.

f’(x0) = 0 at some x0 such that 0 < x0 < 1.

 

Solution: The answer is E

 

The true statement is precisely the conclusion of Rolle’s Theorem. Which says that given a function f that is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) with f(a) = f(b), then there is some number c in (a,b) such that f’(c) = 0. Hence, the only true statement above is, statement E.