Consider the function $$f(x,y) = e^{-6 x - x^2 + 2 y - y^2}.$$ Find and classify all critical points of the function. If there are more blanks than critical points, leave the remaining entries blank.

$f_x =$
$f_y =$
$f_{xx} =$
$f_{xy} =$
$f_{yy} =$

The critical point with the smallest $x$-coordinate is
(, ) Classification: local minimum local maximum saddle point cannot be determined (local minimum, local maximum, saddle point, cannot be determined)

The critical point with the next smallest $x$-coordinate is
(, ) Classification: local minimum local maximum saddle point cannot be determined (local minimum, local maximum, saddle point, cannot be determined)

The critical point with the next smallest $x$-coordinate is
(, ) Classification: local minimum local maximum saddle point cannot be determined (local minimum, local maximum, saddle point, cannot be determined)