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) ?
Best Answer
Note: Except some technicality issues the following example gives a good intuition behind the mean value theorems.
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 (which is $c$ seconds) 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!