Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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) ?

share|cite|improve this question
I think the tag of calculus is more appropriate for this question, rather than real analysis. – awllower Feb 6 '13 at 10:02
See here… for the theorem together with a proof – k1next Feb 6 '13 at 10:04
Possible duplicate – awllower Feb 6 '13 at 10:27
up vote 33 down vote accepted

In the 2012 Olympics Usain Bolt won the 100 metres gold medal with a time of 9.63 seconds.
His average speed was total distance, $d(t_2)-d(t_1)$, over total time, $t_2-t_1$: $$V_a=\dfrac{d(t_2)-d(t_1)}{t_2-t_1}=\dfrac{100}{9.63}=10.384 \ \text{m/s}=37.38 \ \text{km/h}.$$ Mean value theorem $$ f'(c)=\dfrac{f(b)-f(a)}{b-a} $$ says that at some point Bolt was actually running at the average speed of $37.38$ km/h.

Powell Asafa was participating in that race also, with a time $11.99=1.245\times9.63$ seconds, so Bolt's average speed was $1.245$ times the average speed of Powell. Generalized mean value theorem: $$\dfrac{f'(c)}{g'(c)}=\dfrac{f(b)-f(a)}{g(b)-g(a)}=\dfrac{\frac{f(b)-f(a)}{b-a}}{\frac{g(b)-g(a)}{b-a}},$$ says that at some point Bolt was actually running at a speed exactly $1.245$ times of Powell's speed!

share|cite|improve this answer
I know you must think that a proof is very easy. But per chance it is better to include a proof? Thanks in any case. – awllower Feb 6 '13 at 11:00
This is what I meant by intuition. Thanks – Applied mathematician Feb 6 '13 at 12:26
Best example I've seen yet. Thank you @P.. – Mr.Fry Jul 10 '14 at 0:23

Consider the function:
Then clearly $h(b)-h(a)=0$. Hence, by the ordinary mean-value theorem, $h'(c)=0$ for some $c$ in between $b$ and $a$. And that is what we wanted.
As for the intuition, it could be thought of as requiring a tangent line to a parameterised curve on the plane, as in the Wiki-article. Also it is in fact equivalent to the ordinary mean-value theorem, as the proof shows.
P.S. Suppose $g'(x) \neq 0$, then there is an inverse of $g$. If we substitute $y=g^{-1}(x)$, then the theorem is nothing but the ordinary mean-value theorem. Thus, in some sense, this "generalised" mean-value theorem could even be deemed as a "specialized" version of mean-value theorem.

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.