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 ∈ R with a < b and 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 (b) − f (a) f ′ (c)= . (4.3) b− a Proof: The linear function whose graph goes through (a, f (a)) and (b, f (b)) is f (b) − f (a) g(x)= (x − a)+ f (a). b − a Defne f (b) − f (a) h(x)= f (x) − g(x)= f (x) − (x − a)+ f (a) for x ∈ [a,b]. b − a Then h(a)= h(b), and h satisfes the assumptions of Theorem 4.2.2. Thus, there exists c ∈ (a,b)
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