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:
