In the calculus of variations (unconstrained), one applies a first-order variation on a general functional of the form

$$\int\_{a}^{b}F(x,y,y')\,dx$$

to obtain the first-order necessary condition for optimality as expressed by the Euler-Lagrange equality

$$\frac{d}{dx}F\_{y'} -F\_y = 0.$$

What is the necessary condition for a minima? How about a sufficient condition?