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!
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: $(2,2)$