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Let $f,g:[a,b] \rightarrow \mathbb{R}$ continuous functions and differentiable in $(a,b)$ , show that $\exists c \in (a,b)$ such $[f(b)-f(a)]g'(c) = [g(b)-g(a)]f'(c)$

I tried using the mean value theorem, but I can not relate the two functions with the same c.

any help is appreciated

Ricardo Gomes
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1 Answers1

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This is the Cauchy Mean Value Theorem.

Hint: Work with

$$F(x) = \left[ f(b) - f(a) \right] g(x) - \left[g(b) - g(a) \right] f(x) $$

on the interval $(a, b)$. Evaluate $F(a)$ and $F(b)$. After expanding fully, what do you notice? An application of Rolle's theorem will complete the proof.

MathMajor
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  • I am not that familiar with the custom practice here on MSE, but I pointed out a possible duplicate which has an answer with a quite similar proof. So why not just flag/vote to close as a duplicate? – Martin R Dec 31 '14 at 22:31
  • I looked at the link and the answer given did not provide a proof ... it only gave an example and the intuition behind the theorem. – MathMajor Dec 31 '14 at 22:33
  • Did you have a look at the second answer http://math.stackexchange.com/a/296187/42969 ? – Martin R Dec 31 '14 at 22:34
  • No I did not. I see it now. I'll leave this answer here unless the OP indicates that I should remove it. I also approached it in a different manner using Rolle's Theorem instead of IVT (yes I know they are similar). – MathMajor Dec 31 '14 at 22:35