DescriptionLagrange simple.svg	English: derivative of https://en.wikipedia.org/wiki/File:Lagrange_simple.jpg
Date	7 August 2012
Source	Own work based on: Lagrange simple.svg en:File:Lagrange simple.jpg
Author	Original: Jacobmelgaard   Vector: Zerodamage
SVG development InfoField	The SVG code is valid.   This oversized trigonometry was created with Inkscape.    This trigonometry uses embedded text that can be easily translated using a text editor.

The graph was made in Matlab with the following source code

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

% Constraint data
t = -pi:0.01:pi;
xc = sqrt(3)*cos(t);
yc = sqrt(3)*sin(t);
zc = xc.^2 .* yc;

% Plot graph and constraint
figure();
surface = surfc(X,Y,Z);

hold on;
plot3(xc, yc, zc, 'k', 'LineWidth', 2);
plot3(xc, yc ,-10*ones(1,length(t)), 'k', 'LineWidth', 1);
set(gcf, 'Color', [1 1 1]);
set(gcf, 'Position', [46 107 767 682]);
% Axis labels
text(0.2, -3, -10, 'x');
text(2.5, -.5, -10, 'y');
text(-2.4, -2, 0, 'z');

% Annotations of the max and min points
% Point indicating max #1
plot3(sqrt(2), 1, 2, 'kx', 'MarkerSize', 14, 'LineWidth', 2);
plot3(sqrt(2), 1, -10, 'kx', 'MarkerSize', 10, 'LineWidth', 2);
text('Interpreter', 'latex',...
        'String', '$$(\sqrt{2}, 1, 2)$$',...
        'Position', [1.1 .1 2.1],...
        'FontSize', 20);
% Point indicating max #2
plot3(-sqrt(2), 1, 2, 'kx', 'MarkerSize', 14, 'LineWidth', 2);
plot3(-sqrt(2), 1, -10, 'kx', 'MarkerSize', 10, 'LineWidth', 2);
text('Interpreter', 'latex',...
        'String', '$$(-\sqrt{2}, 1, 2)$$',...
        'Position', [-1.6 .1 2.1],...
        'FontSize', 20);
% Point indicating min #1
plot3(sqrt(2), -1, -2, 'kx', 'MarkerSize', 14, 'LineWidth', 2);
plot3(sqrt(2), -1, -10, 'kx', 'MarkerSize', 10, 'LineWidth', 2);
plot3(0,1,-3,'kx')
text('Interpreter', 'latex',...
        'String', '$$(\sqrt{2}, -1, -2)$$',...
        'Position', [.5 -.8 -1.4],...
        'FontSize', 20);
% Point indicating min #2
plot3(-sqrt(2), -1, -2, 'kx', 'MarkerSize', 14, 'LineWidth', 2);
plot3(-sqrt(2), -1, -10, 'kx', 'MarkerSize', 10, 'LineWidth', 2);
text('Interpreter', 'latex',...
        'String', '$$(-\sqrt{2}, -1, -2)$$',...
        'Position', [-1.9 -.8 -1],...
        'FontSize', 20);
hold off;
shading interp;
view([25 18]);
%plot2svg must be retrieved from http://www.zhinst.com/blogs/schwizer/
plot2svg;

Corresponding Mathematica code:

Block[{x, y, f = #^2 #2 &, g = #^2 + #2^2 &, c = 3},
 sol = With[{max = MaxValue[{f[x, y], g[x, y] == c}, {x, y}]},
   Solve[{f[x, y] == max, g[x, y] == c}, {x, y}, Reals]
   ];
 Show[
  Plot3D[f[x, y], {x, -2, 2}, {y, -2, 2}, Mesh -> {{c}}, 
   MeshFunctions -> g, PlotRange -> All, Boxed -> False, 
   AspectRatio -> 1, PlotPoints -> 30, MeshStyle -> Cyan, 
   PerformanceGoal -> "Quality", ColorFunction -> "DeepSeaColors"],
  Graphics3D[{Red, PointSize[Large], Point[{x, y, f[x, y]} /. sol]}]
  ]
 ]

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