 Legendre necessary condition

A nonnegative second variation requires its highest-derivative coefficient $P=f_{vv}$ to be nonnegative. If it were negative on an interval, a smooth compactly supported factor times $\sin (nx)$ would have negative derivative energy of order $n^2$, while its zero-order energy stays bounded. Large $n$ would contradict nonnegativity. This is a local necessary condition; it does not rule out negative directions caused by the lower-order terms.