97 Figure 4.2: Illustration of Rolle’s Theorem. Figure 4.3: Illustration of the Mean Value Theorem. Theorem 4.2.3 — Mean Value Theorem. Let a,b∈Rwith a<band f : [a,b] →R. Suppose f is continuous on [a,b] and differentiable on (a,b). Then there exists c ∈(a,b) such that f ′(c) = f (b)−f (a) b−a . (4.4) Proof: The linear function whose graph goes through(a, f (a)) and(b, f (b)) is g(x) = f (b)−f (a) b−a (x−a)+f (a). Define h(x) =f (x)−g(x) =f (x)− f (b)−f (a) b−a (x−a)+f (a) for x ∈[a,b]. Then h(a) =h(b), and h satisfies the assumptions of Theorem 4.2.2. Thus, there exists c ∈(a,b)
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