# Find and classify all the critical points for f(x,y)=3x^2y-y^3-3x^2+2

Find and classify all the critical points for $f\left(x,y\right)=3{x}^{2}y-{y}^{3}-3{x}^{2}+2$
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Laith Petty

Let z=f(x). Differentiating wrt x and y:
$\frac{dz}{dx}=6xy+3{x}^{2}\frac{dy}{dx}-3{y}^{2}\frac{dy}{dx}-6x,$
$\frac{dz}{dy}=6xy\frac{dx}{dy}+3{x}^{2}-3{y}^{2}-6x\frac{dx}{dy}.$
$\frac{dz}{dx}=3\left({x}^{2}-{y}^{2}\right)\frac{dy}{dx}+6x\left(y-1\right),$
$\frac{dz}{dy}=6x\left(y-1\right)\frac{dx}{dy}+3\left({x}^{2}-{y}^{2}\right).$
The general form is: $\frac{dz}{dt}=6xy\frac{dx}{dt}+3{x}^{2}\frac{dy}{dt}-3{y}^{2}\frac{dy}{dt}-6x\frac{dx}{dt}$, where t is a parameter such that x=g(t), y=h(t) and z=j(t), where g, h and j are functions of t.
When $\frac{dz}{dx}=0$ or $\frac{dz}{dy}=0$ there is a critical point: $\frac{dy}{dx}=2x\frac{y-1}{{y}^{2}-{x}^{2}}.$
The general form is $\frac{dz}{dt}=6x\left(y-1\right)\frac{dx}{dt}+3\left({x}^{2}-{y}^{2}\right)\frac{dy}{dt}=0.$
So x=y=1 is a critical point, x=y=0 is another, x=-1 and y=1 is another.
When x=y=1+d, where d is very small, $\frac{dz}{dt}=6d\left(1+d\right)\frac{dx}{dt}+0$ which is positive when d and dx/dt are both positive or both negative. This suggests a minimum at x=y=1. If x=y=d,