Cauchy's mean-value theorem is a generalization of the usual mean-value theorem. It states that if and are continuous on the closed interval , if , and if both functions are differentiable on the open interval , then there exists at least one with such that

(Hille 1964, p. 340).

Hille, E. Analysis, Vol. 1. New York: Blaisdell, 1964.

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Weisstein, Eric W. "Cauchy's Mean-Value Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CauchysMean-ValueTheorem.html