Mean Value Theorem Integrals

Let $f,g:[a,b]\to\mathbb{R}$ be smooth and integrable. Then, there exists an $x_0\in[a,b]$ with

$$ \int_a^b f(x)g(x)dx=f(x_0)\cdot\int_a^b g(x)dx.$$

Is there any way of approximating $x_0$, without evaluating $\int_a^b f(x)g(x)dx$?

We may assume $g$ to be positive and monotone increasing and $\int_a^b g(x)dx$ to be known.

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