 Resonant sine modes of a quadratic variational integral

The Euler-Lagrange equation is $y^{\prime \prime }+ky=0$. For positive $k$ a nonzero stationary function exists exactly when $\sqrt{k}$ is a positive integer $n$, and all such functions are $B\sin (nx)$. These values are the Dirichlet eigenvalues of $-d^2/d{x}^2$ on the interval. At $k=0$ the solutions are affine and the endpoint conditions leave only zero; negative $k$ likewise leaves only zero. Stationarity alone does not assert minimization of the quadratic functional.