Minimize $\int\limits _{0}^{1} f^2(x) dx$ if $\int\limits_{0}^{1} f(x) dx = 1$, $\int\limits_{0}^{1} xf(x)dx = 1$
What's the minimum of $\int_0^1 f(x)^2 \: dx$, subject to $\int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1$?