A mean value theorem involving two functions

Question

Let $f,g:[a,b] \rightarrow \mathbb{R}$ be continuous in $[a,b]$ and differentiable in $(a,b)$. Prove that there is a point $c \in (a,b)$ such that: $$[f(b)-f(a)]g'(c) = [g(b)-g(a)]f'(c).$$

I think this is a straightforward application of the mean value theorem to a clever function. Am I right? If so, which function? Thanks :)

This seems to be a duplicate question...the previous question was titled Generalized Mean Value Theorem. — lulu, Jul 06 '15 at 00:19
This is definitely a duplicate. Just consider the function $$h(x)=[f(b)-f(a)]g(x)-[g(b)-g(a)]f(x)$$ and apply Rolle's theorem to $h$ on the interval $[a,b]$. — shalin, Jul 06 '15 at 00:21
This is Cauchy's mean value theorem as @vadim123 said, and is crucial for proving L'Hopital's rule - although few calculus students seem to know this. — zhw., Jul 06 '15 at 01:54

Eugene Zhang · Accepted Answer · 2015-07-06T01:51:40.590

3

Let $$ h(x)=(f(x)-f(a))(g(b)-g(a))-(f(b)-f(a))(g(x)-g(a)) $$ Then $h(a)=h(b)=0$. By Rolle's Theorem, there exists $c\in [a,b]$ such that $$ h'(c)=f'(c)(g(b)-g(a))-(f(b)-f(a))g'(c)=0 $$ Or $$f'(c)(g(b)-g(a))=(f(b)-f(a))g'(c)$$

edited Jul 06 '15 at 01:51

answered Jul 06 '15 at 01:45

Eugene Zhang

17,100

A mean value theorem involving two functions

1 Answers1