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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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