Past exam of the calculus of variations course of the University of Oxford / 2011 / 3 / b / ii

The continuous representative of $u$ makes $x\mapsto 2xu(x{)}^3$ continuous on the closed interval. The integrated formula therefore gives a continuously differentiable representative of $u^{\prime }$, and hence $u\in C^2([0,1])$ withDifferentiate once more. The right side is continuously differentiable because $u\in C^2$, andThus $\boxed{u\in C^3([0,1])}$. These formulas give continuous one-sided derivatives at the endpoints as well. Repeated bootstrapping would actually give smoothness of every order.