= Strengthened Legendre condition
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{title2=$P(x)=f_{vv}(x,u,u')>0$}
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>: $\int\varphi^2$ is strictly positive on nonzero variations even though its <derivative> coefficient vanishes.
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