Functions $f(x)$ and $g(x)$ are continuous on the interval $[a,b]$.
Prove that there exist a point $\delta \in (a,b)$ such that $$ f(\delta)\int_{a}^{\delta} g(x) \: dx = g(\delta)\int_{\delta}^{b}f(x) \: dx $$
Comment: I suspect this question requires the Mean Value Theorem, and a suitable change of variable, but I am not able to figure this out.
Let $\varphi(x)=\left(\displaystyle\int_{a}^{x}g(t)dt\right)\left(\displaystyle\int_{x}^{b}f(t)dt\right)$, then $\varphi(a)=\varphi(b)=0$, so $\varphi'(\delta)=0$ for some $\delta\in(a,b)$.