Let $f,g:[a,b]\to\mathbb{R}$ be Riemann integrable functions such that $g$ is monotone. Show that there exists $x_0\in [a,b]$ such that $\int_a^b f(x)g(x)\ dx=g(a)\int_a^{x_0}f(x)\ dx+g(b)\int_{x_0}^bf(x)dx$.
I don't really know how to start with this thing. There is a hint: show that if $g$ is decreasing on $[a,b]$ with $g(b)=0$, then $g(a)\inf_{x\in[a,b]}\int_a^xf(t)\ dt\leq \int_a^bf(x)g(x)dx\leq g(a)\sup_{x\in[a,b]}\int_a^xf(t)\ dt$. Do we need to somehow use mean value theorem? The thing is, I can't imagine what $f(x)g(x)$ would look like without any condition of $f$.
