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