Sign up ×
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.

Suppose $a$ and $b$ from $\mathbb{R}$ as $a<b$ and $f$ and $g$ two continuous function on $[a;b]$ and derivable on $]a;b[$ as $\forall$ $x$ $\in$ $]a;b[$ $g{'}(x) \neq 0$.

How can I prove that $\exists$ $c$ $\in$ $]a;b[\;\;$ s.t. $\;\;\dfrac { f(b)-f(a) }{ g(b)-g(a) } =\dfrac { f^{ ' }\left( c \right) }{ g^{ ' }\left( c \right) } $ using Rolle's theorem.

share|cite|improve this question
By the way, this has a standard name, the Cauchy Mean Value Theorem. – André Nicolas Nov 29 '12 at 22:32

2 Answers 2

up vote 5 down vote accepted


Define the function $$ h(x)=f(x)(g(b)-g(a))-g(x)(f(b)-f(a)) $$ then $h(b)=h(a)$. Apply Rolle's Theorem.

share|cite|improve this answer
+1 and $7$ seconds difference :) – user17762 Nov 29 '12 at 22:11

Look at the function $$h(x) = (f(b) - f(a))g(x) - (g(b) - g(a)) f(x)$$

share|cite|improve this answer
I like this answer, too :-) – robjohn Nov 29 '12 at 22:13

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.