I need help finding critical values of my function

My function is: Could you please hlp me find code to solve for its critical points? Whenever I put it into matlab its unable to find the points.

f(x,y)= (100/((3*x+13)^2+(3*y+2)^2+(13+2))) - ((2*100)/((3*x+5)^2+(3*y-5)^2+10))-((3*100)/((3*x+13)^2+(3*y-5)^2+18))+((4*100)/((3*x-5)^2+(3*y-2)^2+7));

[Stephan]

What ist the different aspect to your last question?

[Meslech Bellay]

i changed the function a bit

[Stephan]

Does this change the approach i gave you in my answer?

Answer 1

[Stephan]

Hi,

use fsolve on the derivates of your function. Then give an educated guess for the initial value of every extremum you are interested in:

% This part is only needed to plot the function
syms x y 
a= 13;
b= 5;
c=2;
d=5;
p = 4*(a+b+c+d);
f(x,y)= (p./((3*x+a).^2+(3*y+c).^2+(a+c)))...
-(2*p./((3*x-b).^2+(3*y+d).^2+(b+d)))...
-(3*p./((3*x+a).^2+(3*y-d).^2+(a+d)))...
+(4*p./((3*x-b).^2+(3*y-c).^2+(b+c)));

% The interesting part starts here
% Find the x,y values for the Maximum
x0_high = [2 1];
sol_high = fsolve(@derivates, x0_high);

% Find the x,y values for the Minimum #1
x0_low_1 = [2 -2];
sol_low_1 = fsolve(@derivates, x0_low_1);

% Find the x,y values for the Minimum #2
x0_low_2 = [-5 2];
sol_low_2 = fsolve(@derivates, x0_low_2);

% Plot function and found extrema. % To see the minima in
% the plot turn it, to see it from the bottom
fsurf(f,[-10,10]);
hold on
colormap(jet); 
% Plot the found point for the Maximum
scatter3(sol_high(1), sol_high(2), f(sol_high(1),sol_high(2)),'ob', 'LineWidth', 3)
% Plot the found point for the Minimum #1
scatter3(sol_low_1(1), sol_low_1(2), f(sol_low_1(1),sol_low_1(2)),'or', 'LineWidth', 3)
% Plot the found point for the Minimum #2
scatter3(sol_low_2(1), sol_low_2(2), f(sol_low_2(1),sol_low_2(2)),'or', 'LineWidth', 3)
hold off

% Define the function that should be solved here. This is the
% hint - fsolve minimizes systems of nonlinear equations of
% the form F(x) = 0 --> Your derivates should be  zero for an
% extremum. Therefore get the dervates, which you can calculate
% without any problems from the symbolic function and get
% numeric functions for your fx and fy. Once you have them,
% replace x by x(1) and y by x(2). This is all i have done,
% to solve the problem. The only trick is giving good guesses
% to the initial values x0 - which can be found by looking at
% the function plot.
function F = derivates(x)
F(1) = 1.0./(x(1).*-1.0e1-x(2).*4.0+x(1).^2.*3.0+x(2).^2.*3.0+1.2e1).^2.*1.0./(x(1).*-1.0e1+x(2).*1.0e1+x(1).^2.*3.0+x(2).^2.*3.0+2.0e1).^2.*1.0./(x(1).*7.8e1+x(2).*1.2e1+x(1).^2.*9.0+x(2).^2.*9.0+1.88e2).^2.*1.0./(x(1).*7.8e1-x(2).*3.0e1+x(1).^2.*9.0+x(2).^2.*9.0+2.12e2).^2.*(x(1).*-1.33528446976e11-x(2).*3.23205188096e12-x(1).^2.*x(2).^2.*1.80827531136e11+x(1).^2.*x(2).^3.*2.91665664e8+x(1).^3.*x(2).^2.*2.10529217664e11+x(1).^2.*x(2).^4.*1.77250032e8+x(1).^3.*x(2).^3.*1.20922932672e11+x(1).^4.*x(2).^2.*1.63437102e11+x(1).^2.*x(2).^5.*9.151784352e9+x(1).^3.*x(2).^4.*2.3286118464e10+x(1).^4.*x(2).^3.*6.7802167152e10+x(1).^5.*x(2).^2.*5.0806762704e10+x(1).^2.*x(2).^6.*1.103321088e9+x(1).^3.*x(2).^5.*1.4486349744e10+x(1).^4.*x(2).^4.*1.1252940228e10+x(1).^5.*x(2).^3.*2.894281632e10+x(1).^6.*x(2).^2.*1.0473694632e10+x(1).^2.*x(2).^7.*7.94860776e8+x(1).^3.*x(2).^6.*8.70478488e8+x(1).^4.*x(2).^5.*6.197538096e9+x(1).^5.*x(2).^4.*2.82083634e9+x(1).^6.*x(2).^3.*8.681812632e9+x(1).^7.*x(2).^2.*2.237073552e9-x(1).^2.*x(2).^8.*5.8970268e7+x(1).^3.*x(2).^7.*4.80055248e8+x(1).^4.*x(2).^6.*2.29884318e8+x(1).^5.*x(2).^5.*1.491262812e9+x(1).^6.*x(2).^4.*6.41482092e8+x(1).^7.*x(2).^3.*1.508295168e9+x(1).^8.*x(2).^2.*5.58426393e8+x(1).^2.*x(2).^9.*7.381125e6-x(1).^3.*x(2).^8.*3.9366e6+x(1).^4.*x(2).^7.*9.073863e7+x(1).^5.*x(2).^6.*6.121413e7+x(1).^6.*x(2).^5.*1.6671501e8+x(1).^7.*x(2).^4.*1.3030146e8+x(1).^8.*x(2).^3.*1.21345695e8+x(1).^9.*x(2).^2.*9.9694395e7-x(1).^2.*x(2).^10.*1.062882e6+x(1).^3.*x(2).^9.*2.95245e5+x(1).^4.*x(2).^8.*2.657205e6+x(1).^5.*x(2).^7.*5.9049e5+x(1).^6.*x(2).^6.*1.062882e7+x(1).^7.*x(2).^5.*5.9049e5+x(1).^8.*x(2).^4.*1.3286025e7+x(1).^9.*x(2).^3.*2.95245e5+x(1).^10.*x(2).^2.*7.440174e6-x(1).*x(2).*1.468880331264e12-x(1).*x(2).^2.*8.28235051776e11+x(1).^2.*x(2).*5.07038485248e11-x(1).*x(2).^3.*4.59252951552e11+x(1).^3.*x(2).*5.40627457536e11-x(1).*x(2).^4.*5.249375424e10+x(1).^4.*x(2).*5.7381994944e10-x(1).*x(2).^5.*3.0201424128e10+x(1).^5.*x(2).*2.403999648e9-x(1).*x(2).^6.*3.8129616e7+x(1).^6.*x(2).*2.4780630672e10-x(1).*x(2).^7.*5.25078288e8+x(1).^7.*x(2).*1.3931388288e10-x(1).*x(2).^8.*1.98290916e8+x(1).^8.*x(2).*3.61130562e9-x(1).*x(2).^9.*8.516178e6+x(1).^9.*x(2).*5.05603782e8-x(1).*x(2).^10.*7.696053e6+x(1).^10.*x(2).*3.1866777e7+x(1).*x(2).^11.*5.9049e4+x(1).^11.*x(2).*5.9049e4+x(1).^2.*5.744419328e11+x(1).^3.*1.15004841984e11-x(1).^4.*2.4410921472e10+x(1).^5.*1.0675027584e10+x(1).^6.*1.9809183744e10+x(1).^7.*6.938244864e9+x(1).^8.*1.366017696e9+x(1).^9.*4.85006616e8+x(1).^10.*1.6559964e8+x(1).^11.*2.6847612e7+x(1).^12.*1.594323e6-x(2).^2.*1.35987270016e12-x(2).^3.*9.512079552e11-x(2).^4.*1.69992437184e11-x(2).^5.*1.14589320192e11-x(2).^6.*1.398593952e10-x(2).^7.*7.99182288e9-x(2).^8.*1.041942204e9-x(2).^9.*3.32170308e8-x(2).^10.*4.0199247e7-x(2).^11.*6.121413e6-x(2).^12.*5.31441e5-2.70806226944e12).*(-8.0e2./3.0);
F(2) = 1.0./(x(1).*-1.0e1-x(2).*4.0+x(1).^2.*3.0+x(2).^2.*3.0+1.2e1).^2.*1.0./(x(1).*-1.0e1+x(2).*1.0e1+x(1).^2.*3.0+x(2).^2.*3.0+2.0e1).^2.*1.0./(x(1).*7.8e1+x(2).*1.2e1+x(1).^2.*9.0+x(2).^2.*9.0+1.88e2).^2.*1.0./(x(1).*7.8e1-x(2).*3.0e1+x(1).^2.*9.0+x(2).^2.*9.0+2.12e2).^2.*(x(1).*-3.238953472e12+x(2).*3.873232654336e12+x(1).^2.*x(2).^2.*1.533112971264e12+x(1).^2.*x(2).^3.*2.317831842816e12+x(1).^3.*x(2).^2.*4.6093833216e10+x(1).^2.*x(2).^4.*1.02211569792e12+x(1).^3.*x(2).^3.*1.008555411456e12+x(1).^4.*x(2).^2.*8.6593520448e10+x(1).^2.*x(2).^5.*1.98512617536e11+x(1).^3.*x(2).^4.*4.39636408704e11+x(1).^4.*x(2).^3.*2.79092739456e11+x(1).^5.*x(2).^2.*1.40617933632e11+x(1).^2.*x(2).^6.*8.964868752e10+x(1).^3.*x(2).^5.*7.0256844288e10+x(1).^4.*x(2).^4.*1.41070166352e11+x(1).^5.*x(2).^3.*5.6303900928e10+x(1).^6.*x(2).^2.*5.3956684224e10+x(1).^2.*x(2).^7.*8.300032416e9+x(1).^3.*x(2).^6.*3.032861616e10+x(1).^4.*x(2).^5.*1.8439979184e10+x(1).^5.*x(2).^4.*3.3225883776e10+x(1).^6.*x(2).^3.*1.3672354176e10+x(1).^7.*x(2).^2.*9.119405088e9+x(1).^2.*x(2).^8.*2.387127996e9+x(1).^3.*x(2).^7.*2.725596864e9+x(1).^4.*x(2).^6.*5.724498744e9+x(1).^5.*x(2).^5.*4.471032816e9+x(1).^6.*x(2).^4.*4.618261656e9+x(1).^7.*x(2).^3.*3.235780224e9+x(1).^8.*x(2).^2.*7.27772364e8+x(1).^2.*x(2).^9.*1.3896198e8+x(1).^3.*x(2).^8.*4.7908422e8+x(1).^4.*x(2).^7.*5.5427328e8+x(1).^5.*x(2).^6.*6.5426292e8+x(1).^6.*x(2).^5.*8.306226e8+x(1).^7.*x(2).^4.*3.503574e8+x(1).^8.*x(2).^3.*5.5348596e8+x(1).^9.*x(2).^2.*2.322594e7+x(1).^2.*x(2).^10.*8.26686e5+x(1).^3.*x(2).^9.*4.251528e7+x(1).^4.*x(2).^8.*1.476225e6+x(1).^5.*x(2).^7.*8.503056e7+x(1).^6.*x(2).^6.*1.18098e6+x(1).^7.*x(2).^5.*8.503056e7+x(1).^8.*x(2).^4.*2.95245e5+x(1).^9.*x(2).^3.*4.251528e7-x(1).^10.*x(2).^2.*1.18098e5+x(1).*x(2).*8.822107136e10+x(1).*x(2).^2.*3.866813955072e12-x(1).^2.*x(2).*1.283635190784e12+x(1).*x(2).^3.*3.505514323968e12-x(1).^3.*x(2).*3.17988744192e11+x(1).*x(2).^4.*1.722868144128e12+x(1).^4.*x(2).*1.08586296576e11+x(1).*x(2).^5.*3.97325523456e11+x(1).^5.*x(2).*5.1682126464e10+x(1).*x(2).^6.*1.88218422144e11+x(1).^6.*x(2).*1.8240520896e10+x(1).*x(2).^7.*2.1997324224e10+x(1).^7.*x(2).*8.044380864e9+x(1).*x(2).^8.*8.413371504e9+x(1).^8.*x(2).*3.109284144e9+x(1).*x(2).^9.*6.17626296e8+x(1).^9.*x(2).*8.72717976e8+x(1).*x(2).^10.*1.26207396e8+x(1).^10.*x(2).*1.38332124e8+x(1).*x(2).^11.*8.503056e6+x(1).^11.*x(2).*8.503056e6+x(1).^2.*7.8152449536e11-x(1).^3.*6.6547643904e11-x(1).^4.*5.0200973568e11-x(1).^5.*8.6134489344e10-x(1).^6.*4.9171536e9-x(1).^7.*7.288335936e9-x(1).^8.*6.050280096e9-x(1).^9.*2.191234032e9-x(1).^10.*3.76365204e8-x(1).^11.*2.5745364e7-x(1).^12.*5.9049e4+x(2).^2.*2.810802945024e12+x(2).^3.*2.931968710656e12+x(2).^4.*1.271715708672e12+x(2).^5.*3.5883603072e11+x(2).^6.*1.57096796544e11+x(2).^7.*1.9837478016e10+x(2).^8.*8.585485488e9+x(2).^9.*4.23123264e8+x(2).^10.*1.7675334e8+x(2).^11.*1.57464e5+x(2).^12.*1.77147e5-7.04921053184e12).*(-2.0e2./3.0);
end

This example gives you:

That way you get the points of interest. Here marked in red (minima) and blue (maximum) for better contrast.

Best regards

Stephan