identifying the critical points of a bivariate function

Dear all,

I have this function below and I plotted it in a 3D space. Is it possible to mark the critical points on the graph so as to be visible to the reader? Similarly for the max or min value of the function?

Best

syms x y
f = 4*x.^2+3*y.^2 ;
   
     
 
    fx = diff(f,x);
    fy = diff(f,y);
 
    [xc,yc] = solve(fx,fy,x,y); [xc,yc];

     fxx = diff(fx,x); fxy = diff(fx,y); fyy = diff(fy,y);
    D = fxx*fyy - fxy^2;

     subs(D,{x,y},{xc(1),yc(1)});
    subs(fxx,{x,y},{xc(1),yc(1)});

 
 
    figure(1); ezsurf(f,[-10,50,-10,50]);

Answer 1

[Vedant Shah]

Hi @ektor,

To mark the critical points on the 3D surface plot and classify whether each is a maximum, minimum, or saddle point, the following approach can be adopted:

Remove the following lines, as they are no longer necessary for evaluating the critical points individually:

subs(D,{x,y},{xc(1),yc(1)}); 
subs(fxx,{x,y},{xc(1),yc(1)}); 

Add the following block of code after plotting the surface using ‘ezsurf’. This section loops through each critical point, evaluates the second derivative test, classifies the point, and plots it with a label using ‘plot3’ function:

hold on 

for k = 1:length(xc) 
    % Evaluate at each critical point 
    Dval = double(subs(D, {x, y}, {xc(k), yc(k)})); 
    fxxval = double(subs(fxx, {x, y}, {xc(k), yc(k)})); 
    zc = double(subs(f, {x, y}, {xc(k), yc(k)})); 
    xc_num = double(xc(k)); 
    yc_num = double(yc(k)); 
    
    % Classify the critical point 
    if Dval > 0 
        if fxxval > 0 
            label = 'Minimum'; 
            color = 'r'; 
        else 
            label = 'Maximum'; 
            color = 'b'; 
        end 
    elseif Dval < 0 
        label = 'Saddle'; 
        color = 'g'; 
    else 
        label = 'Inconclusive'; 
        color = 'k'; 
    end 
    
    % Plot the critical point 
    plot3(xc_num, yc_num, zc, 'o', 'MarkerSize', 10, 'MarkerFaceColor', color) 
    text(xc_num, yc_num, zc, ['  ', label], 'Color', color, 'FontSize', 12) 
end 

hold off 

This enhancement ensures that each critical point is clearly marked on the graph, with its classification visibly labelled.

For more information, refer to the following documentation:

• https://www.mathworks.com/help/matlab/ref/hold.html
• https://www.mathworks.com/help/matlab/ref/plot3.html
• https://www.mathworks.com/help/matlab/ref/text.html