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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.

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

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+1 and $7$ seconds difference :) – user17762 Nov 29 '12 at 22:11

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

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

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