SEEMOUS 2013 Problem 3

Find the maximum value of $$\int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx$$ over all continuously differentiable functions $f:[0,1]\to\mathbb R$ with $f(0)=0$ and $$\int^1_0|f'(x)|^2dx\le1.$$