Cauchy Mean Value Theorem

Cauchy Mean Value Theorem: Let $f,g:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Then there exists $c\in(a,b)$ such that $\bigl(f(b)-f(a)\bigr)g'(c)=\bigl(g(b)-g(a)\bigr)f'(c).$ If moreover $g(b)\neq g(a)$ and $g'(c)\neq 0$, then this can be rewritten as $\frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}.$

This theorem is a standard tool for proving L'Hôpital-type results and for comparing rates of change of two functions.

Proof sketch: Consider $h(x)=\bigl(f(b)-f(a)\bigr)g(x)-\bigl(g(b)-g(a)\bigr)f(x).$ Then $h$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $h(a)=h(b)$. Apply Rolle's theorem to $h$ to obtain $h'(c)=0$ and rearrange.