0
$\begingroup$

Suppose $a$ and $b$ from $\mathbb{R}$ as $a 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.

  • 1
    By the way, this has a standard name, the [Cauchy Mean Value Theorem.](http://en.wikipedia.org/wiki/Mean_value_theorem#Cauchy.27s_mean_value_theorem)2012-11-29

2 Answers 2

5

Hint

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.

5

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

  • 1
    I like this answer, too :-)2012-11-29