TITLE: Generalized mean value theorem
QUESTION [35 upvotes]: I know and understand the mean value theorem. But at the moment I don't have the intuition to understand the generalized mean value theorem

If $f$ and $g$ are continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there exists a point $c\in(a,b)$ where$$[f(b)-f(a)]g'(c)=[g(b)-g(a)]f'(c).$$If $g'$ is never zero on $(a,b)$, then the conclusion can be stated as$$\frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}.$$

What is the intuition? And how can you prove this (generally) ?

REPLY [21 votes]: The example of P.. does not satisfy the conditions of generalized mean value theorem. Let me briefly explain.
$f$ is the position of Bolt while $g$ is the position of Powell. Domains of these functions are time. $f$ is defined for $[0,9.63]$ and $g$ is defined for $[0,11.99]$. So the domains are not equal. In this case Generalized Mean Value Theorem will not work. Think about this unrealistic scenario where Powell has waited for the first 9.63 second where Bolt has finished the race and started running at t=9.63, finishing 100m in the remaining 2.36 seconds. In this case there is no instant at which Bolt was running 1.245 times Powell's speed.