# 2013-06 Inequality on the unit interval

Let $$f : [0, 1] \to \mathbb{R}$$ be a continuously differentiable function with $$f(0) = 0$$ and $$0 < f'(x) \leq 1$$. Prove that $\left( \int_0^1 f(x) dx \right)^2 \geq \int_0^1 [f(x)]^3 dx.$

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