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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! – user 1618033 Feb 11 '13 at 17:14
up vote 7 down vote accepted

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. – user 1618033 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! :-) – user 1618033 Feb 11 '13 at 17:30
2  
@user1709828 Fundamental Theorem of Calculus. – 1015 Feb 11 '13 at 17:46

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