Use Rolles Theorem to prove the Mean Value Theorem. Suppose that f(x) is continuous on
Chapter 4, Problem 45(choose chapter or problem)
Use Rolles Theorem to prove the Mean Value Theorem. Suppose that f(x) is continuous on [a, b] and differentiable on (a, b). Let g(x) be the difference between f(x) and the y-value on the secant line joining (a, f(a)) to (b, f(b)), so g(x) = f(x) f(a) f(b) f(a) b a (x a). (a) Show g(x) on a sketch of f(x). (b) Use Rolles Theorem ( 44) to show that there must be a point c in (a, b) such that g (c)=0. (c) Show that if c is the point in part (b), then f (c) = f(b) f(a) b a .
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