Question

What is Cauchy Mean Value Theorem?

Answer 1

Cauchy's Mean-value Theorem

• Cauchy's theorem is the generalization of the Mean-Value theorem.
• It states that if two functions $f\left(x\right)$ and $g\left(x\right)$ are continuous in the closed interval $\left[a,b\right]$ and if both the functions are differentiable in the open interval $\left(a,b\right)$ and $g\left(a\right)\ne g\left(b\right)$ then there exists a $c$ such that $a<c<b$such that $\frac{f\left(b\right)-f\left(a\right)}{g\left(b\right)-g\left(a\right)}=\frac{f\text{'}\left(c\right)}{g\text{'}\left(c\right)}$where $f\text{'}\left(c\right),g\text{'}\left(c\right)$are derivatives of $f\left(c\right),g\left(c\right)$ respectively.