Question 17
Let $f : [0, 1] \to \mathbb{R}$ be a differentiable function, with integrable derivative on $[0, 1]$, such that $f(1) = 0$. Prove that
$$
\int_{0}^{1} (x f'(x))^2 dx \geq 12 \cdot \left( \int_{0}^{1} x f(x) dx \right)^2 .
$$
$$
\int_{0}^{1} (x f'(x))^2 dx \geq 12 \cdot \left( \int_{0}^{1} x f(x) dx \right)^2 .
$$