Find and classify all the critical points for f(x,y)=3x2y−y3−3x2+2

Question

Find and classify all the critical points for   f(x,y)=3x2y−y3−3x2+2

Laith Petty · Accepted Answer

Let z=f(x). Differentiating wrt x and y:

\frac{d z}{d x} = 6 x y + 3 x^{2} \frac{d y}{d x} - 3 y^{2} \frac{d y}{d x} - 6 x,

\frac{d z}{d y} = 6 x y \frac{d x}{d y} + 3 x^{2} - 3 y^{2} - 6 x \frac{d x}{d y} .

\frac{d z}{d x} = 3 (x^{2} - y^{2}) \frac{d y}{d x} + 6 x (y - 1),

\frac{d z}{d y} = 6 x (y - 1) \frac{d x}{d y} + 3 (x^{2} - y^{2}) .

The general form is:

\frac{d z}{d t} = 6 x y \frac{d x}{d t} + 3 x^{2} \frac{d y}{d t} - 3 y^{2} \frac{d y}{d t} - 6 x \frac{d x}{d t}

, 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{d z}{d x} = 0

or

\frac{d z}{d y} = 0

there is a critical point:

\frac{d y}{d x} = 2 x \frac{y - 1}{y^{2} - x^{2}} .

The general form is

\frac{d z}{d t} = 6 x (y - 1) \frac{d x}{d t} + 3 (x^{2} - y^{2}) \frac{d y}{d t} = 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{d z}{d t} = 6 d (1 + d) \frac{d x}{d t} + 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,

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