So I'm not sure what are my critical values since I end up with multiple x and y values.
My given equation: $z = 3x - x^{3} - 2y^{2} + y^{4}$
Set $z_x = 0$ :
$3-3x^{2} = 0 => 1 - x^{2} = 0 => x^{2} = 1 => x = 1, -1$
Set $z_y = 0$ :
$-4y + 4y^{3} = 0 => -y + y^{3} = 0 => y^{3} - y = 0 => y(y^{2} - 1) = 0 => y = 0, 1, -1$
I'm not sure which pair of values are the critical values, since there are 2 x's and 3 y's.