Legendre necessary condition ($f_{vv}(x,u,u^{\prime })\ge 0$)

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.

Strict positivity of the highest-derivative coefficient on the closed parameter interval is the strengthened Legendre condition. Continuity then gives a uniform positive lower bound and makes the Jacobi equation a regular second-order equation. It permits the Jacobi positivity criterion. It is not necessary for mere strict positivity of a second variation: $\int {\phi }^2$ is strictly positive on nonzero variations even though its derivative coefficient vanishes.