Finding the maximum and minimum of a multivariable function on a domain

Question

I recently asked a similar question to this here:
Find the max and min values of a multivariable function on the boundary of a domain

I thought I understood it and would be able to do questions like the one in my previous post, however, I've run into issues again and I'm not entirely sure where I went wrong.

I have the function: $$f(x,y) = y^3 + 3x^2y – 6x^2 – 6y^2 +2$$

I want to find the maximum and minimum of this function on the boundary of the domain $x^2 + y^2 =1$ using the method of Lagrange multipliers.

I started by finding $f_x = 6xy-12x$ and $f_y=3y^2 +3x^2-12y$

I then set $g = x^2 +y^2$ ($g=1$)

I know that $\nabla f = \lambda \nabla g$ so I set $(6xy-12x,3y^2+3x^2-12y)=\lambda (2x,2y)$

I then set $\lambda 2x=6xy-12x$ and found $\lambda = 3y-6$, I then subbed $\lambda$ into $3y^2 +3x^2-12y = \lambda 2y$ and got $y^2 = x^2$

I then tried to sub this into my objective function (I subbed all $x^2$ values for $y^2$) but I got an equation that I can't factorise to find values for: $4y^3-12y^2+2$ which leads me to believe I've made a mistake somewhere.

I'm not entirely sure where I made the mistake so if anyone could point out to me where I went wrong and show me what I should have done it would really help.

Answer 1

You get 6 points there and the final step is just plugging these into the original function f(x, y).