What function $f(x)$ gives the maximal of $$\frac{\int_0^1 \log f(x)\,dx}{\left(\int_0^1 \int_0^1 \min(x,y) f(x) f(y)\,dx\,dy\right)^{\!1/3}}?$$
I've found $f(x) = \sqrt{e}/x$ gives a high value, and I wonder if there's a nice method finding the actual optimum or giving an upper bound. Not just in this case, but for general functionals.