$f,g$ cont. functions on $[a,b]$ and $g$ is non-negative. then $\exists c \in [a,b]$ such that $\int_{a}^{b} f(x)g(x) dx = f(c) \int_{a}^{b} g(x) dx.$

Question

let $f,g$ be continuous functions on $[a,b]$ and suppose that $g$ is non-negative. then there exists apoint $c \in [a,b]$ such that $\int_{a}^{b} f(x)g(x) dx = f(c) \int_{a}^{b} g(x) dx.$

I have seen a solution similar to my problem in the link below, except that this solution assumes that $g>0,$ but my problem assumes that $g \geq 0$, does my problem contain a mistake? or the result is true also if g is non negative?

Existence of $c$ such that $\int_{a}^{b} f(x)g(x)dx = f(c)\int_{a}^{b} g(x)$

Answer 1

If $g\equiv 0$ then you can take any $c$. Otherwise $\int_a^{b} g(x)dx>0$ and that is enough for the proof you have seen. In that proof the only that could cause problem is division by $\int_a^{b} g(x)dx$].