 Solution

The dominant contributions are $2F(10)\cos (2\pi x)+2F(01)\cos (2\pi y)$. Their extrema occur at $x=0$ or $\frac{1}{2}$ and independently $y=0$ or $\frac{1}{2}$, modulo unit translations. The four candidate positions areWhich is a maximum rather than a minimum or saddle depends on the two signs. Higher Fourier terms sharpen these broad maxima into atomic peaks.