In this problem we prove a special case of the Mean Value Theorem where f(a) = f(b)=0

Chapter 4, Problem 44

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In this problem we prove a special case of the Mean Value Theorem where f(a) = f(b)=0. This special case is called Rolles 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 < c < b, such that f (c)=0. By the Extreme Value Theorem, f has a global maximum and a global minimum on [a, b]. (a) Prove Rolles 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 Rolles Theorem in the case that either the global maximum or the global minimum is not at an endpoint. [Hint: Think about local maxima and minima.]