I need to find the absolute minima and maxima of the function $f(x,y) = 12 x^2 + 12 y^2 - x^3 y^3 -5$ in the region bounded by the disk $x^2 + y^2 \le 1$.
I know that $f(x,y)$ has three critical points in its domain, but only one point of the three, namely $(0,0)$, fits in the region bounded by the disk. This point will be a absolute minimum because $f_{xx}(0,0) >0$ and $f_{yy}(0,0) >0$ and $(0,0)$ is the only minimum in the region bounded by the disk.
Therefore I need to find the other "candidate" points for absolute maximums.
The system of equations for Lagrange Multipliers is the following : $$ \begin{split} -3x^2 y^3 + 24x - 2kx &= 0\\ -3y^2 x^3 + 24y - 2ky &= 0\\ x^2 +y^2 -1 &= 0. \end{split} $$ As you can see, this system of equations is more complex that the typical examples on the internet.What ideas do you have to solve this problems?
Thanks.
Note the function is symmetric in $x,y$ so the answers are expected to be symmetric as well.
Case I. $x = 0$
Then $y = \pm 1$ and you get the points $(0, \pm 1)$.
Case II. $y = 0$
Then $x = \pm 1$ and you get the points $(\pm 1, 0)$.
Case III. $x,y \ne 0$
Then the first equation can be divided by $x$ and the second by $y$. Can you finish this case and individually evaluate the candidates?
UPDATE
Note in case 3, the system becomes $$ \begin{split} -3x y^3 &= 2k - 24\\ -3y x^3 &= 2k - 24\\ x^2 +y^2 &= 1. \end{split} $$ It immediately follows that $xy^3 = yx^3$ and since $x,y\ne 0$ we divide by $xy$ to get $x^2=y^2$. Use the last constraint to solve.