Consider a sufficiently regular bivariate function $J(x,y)$. I would like to maximize $$ J\left(\int_0^T f(t) g(t) \mathrm{d} t, \int_0^T f(t)^2 h(t) \mathrm{d} t \right) $$ in $f$.
If I replace $f(t)$ with $f(t) +\varepsilon \eta(t)$ and take $\frac{\mathrm{d}}{\mathrm{d}\epsilon} \big|_{\varepsilon=0}$, I get the first order condition: $$ \frac{\partial J}{\partial x} \int_0^T g(t) \eta(t) \mathrm{d} t + \frac{\partial J}{\partial y} \int_0^T 2 f(t) h(t) \eta(t) \mathrm{d} t = 0.$$
Is it okay to then move $\frac{\partial J}{\partial x}$ and $\frac{\partial J}{\partial y}$ inside the integrals to get $$ f_*(t) \propto \frac{g(t)}{h(t)}?$$
Finding the proportionality constant might be intractable, but it's quite useful to have the shape of the solution.