Simply parametrize the borders and look for the roots of the gradient in the interior of the function domain.

Parametrize the borders and look for the roots of the gradient in the interior of the function domain.

Moreover, analize the limits in the singularities. I did not make the details, but if the root of $(1+Cx)(1+Dy)$ are in $[0,A]\times[0,B]$, it is possible that an infite limit appears.