Past exam of the calculus of variations course of the University of Oxford / 2011 / 2 / b / i

For any zero-endpoint admissible $u$, use $|\sin u|\le |u|$ and the sharp Poincaré inequality on $(0,1)$:The coefficient is positive, and equality forces $u^{\prime }=0$, hence $u=0$ by its endpoint values. ThusIt is therefore both a strict strong and a strict weak local minimizer.