Editing Mean value theorem (section)

===Proof ===
The proof of Cauchy's mean value theorem is based on the same idea as the proof of the mean value theorem.

Define <math>h(x) = (g(b)-g(a))f(x) - (f(b)-f(a))g(x)</math>, then we easily see <math>h(a)=h(b)=f(a)g(b)-f(b)g(a)</math>.

Since <math>f</math> and <math>g</math> are continuous on <math>[a,b]</math> and differentiable on <math>(a,b)</math>, the same is true for <math>h</math>. All in all, <math>h</math> satisfies the conditions of [[Rolle's theorem]]. Consequently, there is some <math>c</math> in <math>(a,b)</math> for which <math>h'(c) = 0</math>. Now using the definition of <math>h</math> we have:
:<math>0=h'(c)=(g(b)-g(a))f'(c)-(f(b)-f(a))g'(c)</math>
The result easily follows.