Solution
= Solution
The dominant contributions are $2F(10)\cos(2\pi x)+2F(01)\cos(2\pi y)$. Their extrema occur at $x=0$ or $\tfrac12$ and independently $y=0$ or $\tfrac12$, modulo unit translations. The four candidate positions are
$$
(0,0),\quad(\tfrac12,0),\quad(0,\tfrac12),\quad
(\tfrac12,\tfrac12).
$$
Which is a maximum rather than a minimum or saddle depends on the two signs. Higher Fourier terms sharpen these broad maxima into atomic peaks.