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I am trying to prove the following: if $f$ and $g$ are Riemann integrable on $[a,b]$, $g$ is non negative and $f$ is bounded then there exists $c$ such that $$\int_a^b f(x)g(x)dx=m\int_a^c g(x)dx+M\int_c^b g(x)dx$$

Thank you for your help.

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    I suppose $m$ and $M$ are the bounds for $f$, so $m\leq|f|\leq M$. Am I correct? – 2017-02-12
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    @flytothesurface yes. Thats correct. I tried to start my proof from assumption that mg(x)<=f(x)g(x)<=Mg(x). Then to take the integral of both sides – 2017-02-12

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Define $ h(y) := \int_a^y \left(f(x) - m \right) g(x) \, dx - \int_y^b \left( M - f(x) \right) g(x) \, dx $ on $[a,b] $.

Then $h$ is continuous and $ h(a) \le 0 $ while $ h (b) \ge 0 $. Apply the intermediate value theorem to get $c$ such that $h(c) = 0 $, simplify and conclude.

You may note that $h$ is suggested by taking the difference between the sides of the equality you have to establish.

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    could you please clarify how did you get this result as the difference between two sides of the equality? – 2017-02-12
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    @Polly: Write the left side as $ \int_a^c + \int_c^b $. – 2017-02-13