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The Legendre test (as mentioned in An Introduction to the Calculus of Variations by Charles Fox, requires that the sign of $\partial^2 F \over\partial y'^2$ is constant throughout the range of integration then (provided that the other 2 conditions are satisfied, the functional of the stationary path is max or min depending on the sign of $\partial^2 F \over\partial y'^2$. What if $\partial^2 F \over\partial y'^2$ vanishes at some points but have the same sign for all others in the range?

Added: The Legendre test: If 1) Euler's equation is satisfied, 2) the range of integration is sufficiently small, 3) sign of $\partial^2 F \over\partial y'^2$ is constant on range of integration Then the functinal of the stationary path is max or min depending on the sign of $\partial^2 F \over\partial y'^2$.

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