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.