extended mean-value theorem

Let $f:[a,b]\to ℝ$ and $g:[a,b]\to ℝ$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Then there exists some number $\xi \in (a,b)$ satisfying:

$$(f(b)-f(a))g^{\prime }(\xi )=(g(b)-g(a))f^{\prime }(\xi ).$$

If $g$ is linear this becomes the usual mean-value theorem.

Title	extended mean-value theorem
Canonical name	ExtendedMeanvalueTheorem
Date of creation	2013-03-22 13:04:11
Last modified on	2013-03-22 13:04:11
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	9
Author	mathwizard (128)
Entry type	Theorem
Classification	msc 26A06
Synonym	Cauchy’s mean value theorem
Synonym	extended mean value theorem
Synonym	generalized mean value theorem
Related topic	MeanValueTheorem