Use rolles theorem to prove the mean value theorem suppose

Question

Question: Use Rolle's 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) (x - a)
                             b - a

(a) Show g(x) on a sketch of f(x).

(b) Use Rolle's Theorem (Problem 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