Let $f:[0,1]\to\mathbb R$ be a continuously differentiable function such that $f(x_0)=0$ for some $x_0\in[0,1]$. Prove that $$\int^1_0f(x)^2dx\le4\int^1_0f’(x)^2dx.$$
Source: 2001 Moldova MO Grade 12 P7
Tags: calculus, integration, inequalities
Let $f:[0,1]\to\mathbb R$ be a continuously differentiable function such that $f(x_0)=0$ for some $x_0\in[0,1]$. Prove that $$\int^1_0f(x)^2dx\le4\int^1_0f’(x)^2dx.$$