# Prove that $\int_a^cf(x)\mathrm{d}x+(c-a)g(c)=\int_c^bg(x)\mathrm{d}x+(b-c)f(c)$

Let $f$ , $g$ be real continuous functions in $[a,b]$. Prove that there is $c\in(a,b)$ such that

$$\int_a^cf(x)\mathrm{d}x+(c-a)g(c)=\int_c^bg(x)\mathrm{d}x+(b-c)f(c)$$

What would you suggest me to do here? Thanks.

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@julien: hmmm. I think you're right. I'll check that right now. Thank you! –  Chris's sis Feb 11 '13 at 17:14

## 1 Answer

Let us consider the function $$H(x)=(b-x)\int_a^xf+(x-a)\int_x^bg$$ This is $C^1$ and $H(a)=H(b)=0$.

So by Rolle, there exists $c\in (a,b)$ such that $H'(c)=0$.

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Perfect! Thanks. –  Chris's sis Feb 11 '13 at 17:25
@Chris'ssisterandpals You're welcome. Thanks for the fun question. –  1015 Feb 11 '13 at 17:29
Welcome! Right! Math means fun! :-) –  Chris's sis Feb 11 '13 at 17:30
@user1709828 Fundamental Theorem of Calculus. –  1015 Feb 11 '13 at 17:46