Problem:
Minimize $I(f)$ subject to the constraint $J(f)\leq 0$, where $$I(f)=\int_{x_1}^{x_2}\frac{dx}{f(x)}\tag{$f:[x_1,x_2]\to \mathbb{R}_{\geq 0}$}$$ $$J(f)=\left(\frac{1}{f}\frac{df}{dx}\right)^2+\frac{f^4}{g^2}-C\tag{$g:[x_1,x_2]\to \mathbb{R}_{\geq 0};~C\in\mathbb{R}_{\geq 0}$}$$
How should I approach this? Numerically? Analytically? Any references/books or techniques? Thanks :)