The extended mean-value theorem (Anton 1984, pp. 543-544), also known as the Cauchy mean-value theorem (Anton 1984, pp. 543) and Cauchy's mean-value formula (Apostol 1967, p. 186), can be stated as follows. Let the functions and be differentiable on the open interval and continuous on the closed interval . Then if for any , then there is at least one point such that

# Extended Mean-Value Theorem

## See also

L'Hospital's Rule, Mean-Value Theorem## Explore with Wolfram|Alpha

## References

Anton, H.*Calculus with Analytic Geometry, 2nd ed.*New York: Wiley, 1984.Apostol, T. M.

*Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra.*Waltham, MA: Blaisdell, p. 186 1967.Hille, E.

*Analysis, Vol. 1.*New York: Blaisdell, 1964.

## Referenced on Wolfram|Alpha

Extended Mean-Value Theorem## Cite this as:

Weisstein, Eric W. "Extended Mean-Value Theorem."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/ExtendedMean-ValueTheorem.html