What is
$$\min_{f\in D} \int_{0}^{1} (1+x^2)f(x)^2\mathrm dx,$$ where $D$ is the collection of all continuous real functions from $[0,1]$ such that in the interval $[0,1]$ we have $f(f(x)) = x^2 $.
Because of all those squares I tried to think of a way to use Cauchy-Schwartz. I also considered calculus of variations.
But I am not good at either of those and maybe trying to use them is wrong.