# extended mean-value theorem

Let $f:[a,b]\to\mathbb{R}$ and $g:[a,b]\to\mathbb{R}$ 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 ExtendedMeanvalueTheorem 2013-03-22 13:04:11 2013-03-22 13:04:11 mathwizard (128) mathwizard (128) 9 mathwizard (128) Theorem msc 26A06 Cauchy’s mean value theorem extended mean value theorem generalized mean value theorem MeanValueTheorem