Evaluating maxima and minima of functions of two variables

Four small towns in a rural area wish to pool their resources to build a television station. If the towns are located at the points (-5,0), (1,7), (9,0) and (0,-8) on a rectangular map grid, where units are in miles, at what point S(x,y) should the station be located to minimize the sum of the distances from the towns, by using MATLAB Code and visualize graphically.

[MANOJ VARMA]

Obtain the maximum and minimum values of f (x,y) = x4 + y4 – x2 - y2 + 1.

Answer 1

[DGM]

How would you solve a minimization problem on paper? If you use the symbolic toolbox, it's basically the same.

syms x y

D = sqrt((x+5)^2 + (y-0)^2) + sqrt((x-1)^2 + (y-7)^2) ...
    + sqrt((x-9)^2 + (y-0)^2) + sqrt((x-0)^2 + (y+8)^2);

Dx = diff(D,x);
Dy = diff(D,y);
S = vpasolve([Dx==0, Dy==0],[x y]);
minlocation = double([S.x S.y])

minlocation = 1×2

0.5333 0.0000

fsurf(D,[-10 10 -10 10]); hold on
plot3(S.x,S.y,subs(D,[x y],minlocation),'*w','markersize',10)
colormap(jet)

view(-20,51)

That said, you can get a fair approximation without the symbolic tools.

N = 1000;
x = linspace(0,1,N);
y = linspace(-0.5,0.5,N).';

D = sqrt((x+5).^2 + (y-0).^2) + sqrt((x-1).^2 + (y-7).^2) ...
    + sqrt((x-9).^2 + (y-0).^2) + sqrt((x-0).^2 + (y+8).^2);

[~,idx] = min(D(:));
[r c] = ind2sub([N N],idx);
minlocation = [x(c) y(r)]

minlocation = 1×2

0.5335 0.0005

figure
surf(x,y,D); hold on
plot3(x(c),y(r),D(r,c),'*w','markersize',10)
shading flat
colormap(jet)

view(-20,51)