LANGRANGE'S MEAN VALUE THEOREM

Proof of the Mean Value Theorem  (L.M.V.T)

Assume Rolle's theorem. The equation of the secant -- a straight line -- through points (a, f(a)) and (b, f(b)) is given by

g(x) = f(a) + [(f(b) - f(a)) / (b - a)](x - a).

The line is straight and, by inspection, g(a) = f(a) and g(b) = f(b). Because of this, the difference f - g satisfies the conditions of Rolle's theorem:

(f - g)(a) = f(a) - g(a) = 0 = f(b) - g(b) = (f - g)(b).

We are therefore guaranteed the existence of a point c in (a, b) such that (f - g)'(c) = 0. But

(f - g)'(x) = f '(x) - g'(x) = f '(x) - (f(b) - f(a)) / (b - a).

(f - g)'(c) = 0 is then the same as

f'(c) = (f(b) - f(a)) / (b - a).