Let $f\colon[0,1]\to\mathbb{R}$ be continuously differentiable on $[0,1]$ and satisfy $f(1)=0$. Show that $$\int_0^1|f(x)|^2dx\leq4\int_0^1 x^2|f'(x)|^2dx.$$
Since $f$ is differentiable on $[0,1]$, I can use mean value theorem -- there exists $\xi$ such that $$f'(\xi)=\frac{f(1)-f(0)}{1-0} = -f(0).$$
Unfortunately, I can't proceed further with the solution and have no clue how to use continuity of derivative. Any hints?
