One of my pals asked me to look at this question
Let $f: [0,1] \to \mathbb{R}$ be differentiable. Suppose that $f(0) = 0$ and $0 < f'(x) < 1$ for all $x \in (0, 1)$, where $f'(x)$ is the derivative of $f$. Prove that
$$ \left(\int_{0}^{1}f(x) \ dx \right)^{2} \geq \int_{0}^{1}(f(x))^{3} \ dx$$
I don't really have a clue as to where to start. Any ideas will be appreciated.