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 $y(x)$ which are continuously differentiable for $a\le x\le b$ and satisfy the boundary conditions $y(a)=A$ and $y(b)=B$

find the function for which the functional $J[y]=\int_{a}^{b}F\left(x,y,y'\right)\mathrm{d}{x}$ has a weak extremum.