Question

What is Cauchy's mean value theorem?

Answer 1

Cauchy's mean value theorem:

A generalization of the mean value theorem, Cauchy's mean value theorem is also referred to as the extended mean value theorem.

It states that $f$ is a continuous function in closed interval $\left[a,b\right]$ and differentiable in the open interval $\left(a,b\right)$, then there exists a point $c$ such that $c\in \left(a,b\right)$ and is given by $f\text{'}\left(c\right)=\frac{\left[f\left(b\right)-f\left(a\right)\right]}{b-a}$.

This theorem is defined for two functions $f\left(x\right),g\left(x\right)$ such that they are continuous on $\left[a,b\right]$ and differentiable on $\left(a,b\right)$ and there exists $c$ such that $a<c<b$ and $g\left(a\right)\ne g\left(b\right)$, then, $\frac{f\left(b\right)-f\left(a\right)}{g\left(b\right)-g\left(a\right)}=\frac{f\text{'}\left(c\right)}{g\text{'}\left(c\right)}$