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Extended Mean-Value Theorem

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 f and g be differentiable on the open interval (a,b) and continuous on the closed interval [a,b]. Then if g^'(x)!=0 for any x in (a,b), then there is at least one point c in (a,b) such that

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

See also

L'Hospital's Rule, Mean-Value Theorem

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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.

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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

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