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Prove that if the functions $g,h:[a,b] \rightarrow \mathbb{R}$ are continuous, with $h(x) \geq 0$ for all $x\in[a,b]$, then there is a point $c$ in $(a,b)$ such that $\int_a^bh(x)g(x)dx=g(c)\int_a^bh(x)dx$.

At first I tried to use the Cauchy Mean Value Theorem, by letting $A(x) =\int_a^xh(t)g(t)dt$ and $B(x) =\int_a^xh(t)dt$. Then by CMVT, we have that there exists a point $c$ such that $$\frac{ \int_a^bh(x)g(x)dx }{ \int_a^bh(x)dx}= \frac{h(c)g(c)}{h(c)}=g(c) $$, and the result follows But this only holds if $h(x)$ is never $0$, which we dont have. So is this the wrong way to do this?

pmal
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  • Combine expression in the second line of the prompt to get a better feel of what you are trying to show. – Mr.Fry May 14 '14 at 03:02
  • I don't understand what you are saying Rod? – pmal May 14 '14 at 03:04
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    Actually the extended mean value theorem says $g'(c)[f(b)-f(a)] = f'(c)[g(b)-g(a)]$ with c between a and b, which gives you your result directly. The ratio only applies when one of the derivatives is not 0. – RRL May 14 '14 at 03:38
  • @RRL Well, the extended mean theorem that we proved has $g'(x) \neq 0$ and is stated as a ratio – pmal May 14 '14 at 03:41
  • I can show you the proof in the answer which shows that the cross multiplied form is the primary result -- if you like. – RRL May 14 '14 at 03:44

2 Answers2

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Use the following form of the extended mean value theorem. If $A$ and $B$ are continuous on $[a,b]$ and have derivatives in $(a,b)$ , then there exists $c$ between $a$ and $b$ such that

$$A'(c)[B(b)-B(a)] = B'(c)[A(b)-A(a)]$$

Proof.

Apply Rolle's Theorem to

$$\phi(x) = [A(x)-A(a)](B(b)-B(a)]- [B(x)-B(a)](A(b)-A(a)]$$

where $\phi(a) = \phi(b) = 0$. We get $\phi'(c) = 0$ for some c in the interval.

RRL
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Look at this textbook on page 139. What you're trying to prove is Theorem 3.3.7 (First Mean Value Theorem for Integrals). A proof is there. I'm not sure if you can get CMVT to work on this, but if you're not satisfying all the conditions of a theorem, you can't use it.

http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF

Paul Hurst
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