Multivariable critical points

Question

I have the following function:

$\ f(x,y) = (x^3-y^2)(x-1) $

Expanding the brackets gives:

$\ f(x,y) = x^4 - x^3 - xy^2 + y^2 $

I understand to find the first derivative of both variables and set to zero:

$\ f_x ==> 4x^3 - 3x^2 - y^2 = 0.....[1] $

$\ f_y ==> y - 2xy = 0.....[2] $

$\ y = 2xy.....[3] $

Substituting [3] into [1]:

$\ 4x^3 - 3x^2 - 4x^2y^2 = 0 $

$\ x^2(4x - 3 - y^2) = 0$

I am stuck at this point and can not factorise further to find the critical points. Any hints or clues would be helpful. Thank you.

Answer 1

The second partial derivative is $-2xy+2y$ which gives $-xy+y=y(-x+1)=0$, $y=0$ or $x=1$. By replacing $y=0$ you have $4x^3-3x^2=0$ or $x=1$, $4-3-y^2=0$.