<s>[INST] Global extreme points of function wrt. two variables Hi I am currently working on a problem in theoretical chemistry and am struggling a bit. While discussing Born-Oppenheimer energy functions, I was asked to find all the extreme points and saddle points of the follwing function:

I honestly didn't encounter such a problem before; I tried looking stuff up and found something here:
https://www.massmatics.de/merkzettel/#!217:Globales_Extremum_-_mehrdimensionale_Funktion
(watch out, it's in German)
Basically, what I should do is find the gradient of the function's first derivatives and then construct a linear system of equations. Which shouldn't really be possible as it is a function of grade 4.
Furthermore, I should find the Hesse matrix, but I don't really know if it is also applicable here.
Is there another trick I could use on this function? I find it most peculiar that the terms are arranged in the way they are, maybe there is another, more easy method here?
Thanks in advance!
 [/INST] We will use these extrema in two varaibles notes as a guide.
We have
$$U(x, y) = (x\left(\frac{x^4}{4}-\frac{x^3}{3}-x^2+3\right) \left(\frac{y^4}{4}-\frac{y^3}{3}-y^2+4\right)$$
Finding critical points
$$f_x = \left(x^3-x^2-2 x\right) \left(\frac{y^4}{4}-\frac{y^3}{3}-y^2+4\right) = 0\\ f_y = \left(\frac{x^4}{4}-\frac{x^3}{3}-x^2+3\right) \left(y^3-y^2-2 y\right) = 0$$
From the first, we have $x = -1, 0, 2$ and from the second, we have $y = -1,0,2$ to give us a total of nine critical points
as
$$(x, y) = (-1,-1),(-1,0),(-1,2),(0,-1),(0,0),(0,-2),(2,-1),(2,0),(2,2)$$
The Hessian determinant, $|H(x, y)|$ is given by
$\left|\left(
\begin{array}{cc}
 \left(3 x^2-2 x-2\right) \left(\dfrac{y^4}{4}-\dfrac{y^3}{3}-y^2+4\right) & \left(x^3-x^2-2 x\right) \left(y^3-y^2-2 y\right) \\
 \left(x^3-x^2-2 x\right) \left(y^3-y^2-2 y\right) & \left(\dfrac{x^4}{4}-\dfrac{x^3}{3}-x^2+3\right) \left(3 y^2-2 y-2\right) \\
\end{array}
\right)\right|$
We also have
$$f_{xx} (x, y) = \left(3 x^2-2 x-2\right) \left(\frac{y^4}{4}-\frac{y^3}{3}-y^2+4\right)$$
For the nine critical points, we find

*

*Saddles: $(-1,0), (0, -1), (0, 2), (2, 0)$


*Local Min: $(-1, 1), (-1, 2), (2, -1)$


*Local Max: $(0, 0)$


*Global Min: $(2,2)$
</s>