Simplest Variational Problem

Problem
Let $F \left({x, y, z}\right)$ be a function with continuous first and second (partial) derivatives with respect to all its arguments.

Then among all functions $y \left({x}\right)$ which are continuously differentiable for $a \le x \le b$ and satisfy the boundary conditions $y \left({a}\right) = A$ and $y \left({b}\right) = B$, find the function for which the functional:
 * $\displaystyle J \left[{y}\right] = \int_a^b F \left({x, y, y'}\right) \, \mathrm d x$

has a weak extremum.