Find critical points of a function with two variables

I'm trying to find critical points to this function but it is so long that when i try to run this:

syms x y z

eqns = (120./((3*x+12).^2+(3*y+8).^2+20))-(240./((3*x-9).^2+(3*y+1).^2+10))-(360./((3*x+12).^2+(3*y-1).^2+13))+(480./((3*x-9).^2+(3*y-8).^2+17)); 

dx = diff(eqns,x)== 0 %=(240(18x-54))/((3x-9)^2+(3y+1)^2+10)^2-(480(18x 54))/((3x-9)^2+(3y-8)^2+17)^2+(360(18x+72))/((3x+12)^2+(3y-1)^2+13)^2-(120(18x+72))/((3x+12)^2+(3y+8)^2+20)^2

dy = diff(eqns,y)== 0% =(240(18y+6))/((3x-9)^2+(3y+1)^2+10)^2+(360(18y-6))/((3x+12)^2+(3y-1)^2+13)^2-(480(18y-48))/((3x-9)^2+(3y-8)^2+17)^2-(120(18y+48))/((3x+12)^2+(3y+8)^2+20)^2

[xcr,ycr]=solve(dx,dy); [xcr,ycr]

I get a spinning blue ball that goes forever

the domain is -10<x<10 -10<y<10

[Walter Roberson]

Your code does a solve() on dx and dy but does not define those? And your code calculates hessmatf but does not use that ?

[Ali Mortazavi]

Sorry, updated it and still having the same problem

[Walter Roberson]

eqns would have to be defined before dx or dy

[Ali Mortazavi]

copied it wrong sorry, still same problem

Answer 1

[Walter Roberson]

The key is two-fold:

syms x y real
eqns = (120./((3*x+12).^2+(3*y+8).^2+20))-(240./((3*x-9).^2+(3*y+1).^2+10))-(360./((3*x+12).^2+(3*y-1).^2+13))+(480./((3*x-9).^2+(3*y-8).^2+17)); 
dx = diff(eqns,x);
dy = diff(eqns,y);
sol = vpasolve([dx,dy])

This is fairly fast, and gives sol.x and sol.y with 6 solutions each. You can then

subs(dx,{x,y},{sol.x,sol.y})
subs(dy,{x,y},{sol.x,sol.y})

and see that the derivatives are very close to 0. The exception is at

x = -4.4546625928986563919737615802286
y = -9.5289449194003238929219876399778

which is a point at which the derivatives are steep enough to give numeric problems. This also happens to be the location that my other tests had been finding. It appears that point is a saddle point.

FF = subs(eqns,sol);
fsurf(eqns,[-10 10 -10 10]);
hold on
scatter3(sol.x,sol.y,FF,30,'r+')

You can see two local minima, two local maxima, and 2 saddle points. (I would not be surprised if there were more saddle points that this process does not locate.)