So I'm doing a little calculus of variations on an eigenvalue problem. The goal of this is to find an upper bound for the $\omega_0$ as follows:
$\omega_0^2 \leq \frac{T_0\int_0^a\left|\frac{d\psi}{dx}\right|^2 dx}{\int_0^a \mu(x)\left|\psi(x)\right|^2dx}$ where $\psi(x)$ is any function vanishing at $x=0$ and $x=a$.
How do I minimize the RHS using calculus of variations? I am not super comfortable with variations yet and I need a rough roadmap.