Mean-Value Theorem

Let  be differentiable on the open interval  and continuous on the closed interval . Then there is at least one point  in  such that

The theorem can be generalized to Cauchy's mean-value theorem.

REFERENCES:

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1097-1098, 2000.

Jeffreys, H. and Jeffreys, B. S. "Mean-Value Theorems." §1.13 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 49-50, 1988.