This is from [wikipedia on MVT.]
If $G: [a, b] \mathbb{R}$ is a positive monotonically decreasing function and $\phi : [a, b] \mathbb{R}$ is an integrable function, then there exists a number $x \in (a, b]$ such that$$ \int^b_a G(t)\phi(t)\,dt = G(a^+)\int^x_a\phi(t)\,dt$$Here $G(a^+)$ stands for $lim_{x\rightarrow a+} G(x)$, the existence of which follows from the conditions. Note that it is essential that the interval $(a, b]$ contains $b$.
- I can prove it for the case when $\phi\geq0$. Let $f(x)=\int^x_a\phi(t)\,dt$, then $0\leq f(x)\leq f(b)$. I also have $0\leq\int^b_a 𝐺(𝑡)\phi(t)\,dt\leq G(a^+)f(b)$. As $f(x)$ is continuous, dividing the last equation by $G(a^+)$ yields our desired relation.
- If $\phi$ is integrable, I tried splitting into positive and negative parts $\phi_+$ and $\phi_-$. But did not proceed far? Any hints?