1. The problem statement, all variables and given/known data Suppose that f(x) is a twice-differentiable function defined on the closed interval [a,b]. If f'(c) = 0 for a < c < b, which of the following must be true? I. f(a) = f(b) II. f has a relative extremum at x = c. III. f has a point of inflection at x = c. 2. Relevant equations Rolles Theorem states that there is a c such that f'(c) = 0 between [a,b] if f(x) is continuous on [a,b] and differentiable on (a,b). 3. The attempt at a solution By Rolles Theorem, statement "I" must be correct. (That is, the endpoints are equal to each other). Is this correct? I also put statement "II" as correct because since f'(c) = 0 for a < c < b... c must be a critical point or relative extremum. Question though: does this mean there are other points between a and b where f'(c) = 0 ? For statement "III", I said this was incorrect. It's twice differentiable yes, but we don't know if f(x) has a point of inflection at x = c. Thanks!