= Legendre necessary condition
{c}
{title2=$f_{vv}(x,u,u')\geq0$}
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.
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