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Showing content from https://en.wikipedia.org/wiki/File:Lagrange_very_simple-1b.svg below:

File:Lagrange very simple-1b.svg - Wikipedia

English: Illustration of the constrained optimization problem 1b

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The graph was made in Matlab with the following source code.

% Function data
[X,Y] = meshgrid(-1.2:0.1:1.2, -1.2:0.1:1.2);
Z = (X+Y).^2;

% Constraint data
t = -pi:0.01:pi;
xc = cos(t);
yc = sin(t);
zc = (xc+yc).^2;

% Plot graph and 'constraint'
figure();
surfc(X,Y,Z);
hold on;
plot3(xc, yc, zc, 'k', 'LineWidth', 2);
plot3(xc, yc, 0*ones(1,length(t)), 'k', 'LineWidth', 1);

set(1, 'Color', [1 1 1]);
set(1, 'Position', [46 107 767 682]);
% Axis labels
text(0.0, -2.2, 0, 'x');
text(2.0, -.3, 0, 'y');
text(-1.5, -1.5, 3, 'z');

% Annotations of the max and min points
% Points indicating max
plot3(sqrt(2)/2, sqrt(2)/2, 2, 'kx', 'MarkerSize', 10, 'LineWidth', 1);
plot3(sqrt(2)/2, sqrt(2)/2, 0, 'kx', 'MarkerSize', 10, 'LineWidth', 1);
text('Interpreter', 'latex', 'String', '$$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 2\right)$$',...
        'Position', [+.3 .6 2.4], 'FontSize', 9);

plot3(-sqrt(2)/2, -sqrt(2)/2, 2, 'kx', 'MarkerSize', 10, 'LineWidth', 1);
plot3(-sqrt(2)/2, -sqrt(2)/2, 0, 'kx', 'MarkerSize', 10, 'LineWidth', 1);
text('Interpreter', 'latex', 'String', '$$\left(\frac{-\sqrt{2}}{2}, \frac{-\sqrt{2}}{2}, 2\right)$$',...
        'Position', [-0.9 -0.5 +2.2], 'FontSize', 9);

% Points indicating min
plot3(-sqrt(2)/2, sqrt(2)/2, 0, 'kx', 'MarkerSize', 10, 'LineWidth', 1);
text('Interpreter', 'latex', 'String', '$$\left(\frac{\sqrt{2}}{2}, \frac{-\sqrt{2}}{2}, 2\right)$$',...
        'Position', [+0.8 -1.3 +1.1], 'FontSize', 9, 'Color', 'white');

plot3(sqrt(2)/2, -sqrt(2)/2, 0, 'kx', 'MarkerSize', 10, 'LineWidth', 1);
text('Interpreter', 'latex', 'String', '$$\left(\frac{-\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 2\right)$$',...
        'Position', [-0.2 -0.5 +1.0], 'FontSize', 9, 'Color', 'white');

hold off

view([34 24]);

%interpolate for improved rendering
shading interp;

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