$$ f(x,y)= x^3 + 2xy - y^2 $$
$$\frac{\partial f(x,y)}{\partial x}= 3\, x^2 + 2y $$
$$\frac{\partial f(x,y)}{\partial y}= 2x - 2y$$
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$$0 = 3\, x^2 + 2y $$
$$0 = 2x - 2y$$
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$$x =y$$
$$0= 3\, x^2 + 2x $$
$$0= x \cdot (3\, x + 2 ) $$
$$x_1=0 $$
$$0=3\, x + 2 $$
$$x_2=-\frac 23\, $$
$$y_1=0 $$
$$y_2=-\frac 23\, $$
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$$K_1=(0,0) $$
$$K_2=(0,-\frac 23)\, $$
$$K_3=(-\frac 23,0)\, $$
$$K_4=(-\frac 23,-\frac 23)\, $$
