Proving the integral of $f(x)g(x)$ on [a,b] is equal to an expression

Question

Given a continuous function g and an increasing function f on $[a,b]$, I'm trying to prove that
$\int_{a}^{b}f(x)g(x) dx = f(a)\int_{a}^{c} g(x)dx + f(b)\int_{c}^{b}g(x)dx$ for some c in $[a,b]$.
I tried to proceed by looking at the function $u(x)=f(a)\int_{a}^{x}g(t)dt+f(b)\int_{x}^{b}g(t)dt$, which is obviously continuous. I already know that $u(x)$ can achieve its maximum and minimum on $[a,b]$. If somehow I can prove that $\int_{a}^{b}f(x)g(x) dx$ is in $[{min}_{x\in [a,b]}(u(x)),{max}_{x\in [a,b]}(u(x))]$ then the proof can be easily done, but I really have no clue on how to prove this step. Any thoughts on that?

Answer 1

Hint: use that $f$ is an increasing function to find $\min_{x\in[a,b]}u(x)$ and $\max_{x\in[a,b]}u(x)$.