The Mean Value Theorem

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The mean value theorem refers to the mean or average rate of the change of f in the interval [a,b]. Listed below are two variations of the mean value theorem. The first is the basic theorem and the second is an extended version.

The Mean Value Theorem
If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that f'(c) = (f(b) - f(a)) / (b - a)

The Extended Mean Value Theorem
If f and g are differentiable on the open interval (a, b) and continuous on [a b] such that g'(x) does not = 0 for any x in (a, b), then there exists a point c in (a, b) such that f'(c)/g'(c) = (f(b) - f(a)) / (g(b) - g(a))