Question: In this problem we prove a special case of the Mean Value Theorem where f(a) = f(b) = 0. This special case is called Rolle's Theorem: If f is continuous on [a, b] and differentiable on (a, b), and if f(a) = f(b) = 0, then there is a number c, with a
f'(c) = 0
By the Extreme Value Theorem, f has a global maximum and a global minimum on [a, b].
(a) Prove Rolle's Theorem in the case that both the global maximum and the global minimum are at endpoints of [a, b]. [Hint: f(x) must be a very simple function in this case.]
(b) Prove Rolle's Theorem in the case that either the global maximum or the global minimum is not at an endpoint.