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Lagrange multiplier
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<iframe 
    src="http://jsxgraph.uni-bayreuth.de/share/iframe/lagrange-multiplier" 
    style="border: 1px solid black; overflow: hidden; width: 550px; aspect-ratio: 55 / 65;" 
    name="JSXGraph example: Lagrange multiplier" 
    allowfullscreen
></iframe>
This code has to 
    <div id="divexamples">
        <strong>Objective functions:</strong><br />
        <select id="examples" onchange="setText();" style="max-width: 600px; font-size: 80%">
            <option>x**2 - 6*x + y**2 - 4*y + 13</option>
            <option>x^2+y^2+x*y*2-2</option>
            <option>x^2 + y^2 + 2*x*y</option>
            <option selected="selected">sin(PI*(x**2 + y)) + cos(PI*y)</option>
            <option>1 / 9 * x ** 2 + y ** 2 - 1</option>
        </select><br />
        <strong>Constraints (function equals 0):</strong><br />
        <select id="examples2" onchange="setText2();" style="max-width: 600px; font-size: 80%">
            <option>-x^2-y+1</option>
            <option selected="selected">x^2+y-1</option>
            <option>-y+1</option>
            <option>y-0.1</option>
            <option>y-0.001</option>
            <option>4*(x-7)**2 + 4*(y-4)**2 - 5</option>
            <option>-x+y</option>
            <option>x + y - 2</option>
            <option>x^2 + y^2 + 2*x*y</option>
            <option>1 / 16 * x ** 2 + y ** 2 - 1</option>
            <option>x</option>
        </select>
    </div>
    <div id="divequation">
        <strong>Equation:</strong><br />

        0 =
        <input id="function" type="text" value="sin(PI*(x**2 + y)) + cos(PI*y)" style="font-size: 80%; width: 480px" />
        <button onClick="setObjective();plot();">select</button><br />
        0 =
        <input id="function2" type="text" value="x^2+y-1" style="font-size: 80%; width: 480px" />
        <button onClick="setConstraint();plot();">select</button><br />
        <strong>Kontourlinien</strong><br />
        <input id="niveauline" type="text" value="[-1.5,-0.5,0,0.5,1.5]" style="font-size: 80%; width: 480px" />
        <button onClick="setContour();plot();">set</button><br />

    </div>

<div id="board-0-wrapper" class="jxgbox-wrapper " style="width: 100%; ">
   <div id="board-0" class="jxgbox" style="aspect-ratio: 1 / 1; width: 100%;" data-ar="1 / 1"></div>
</div>

<script type = "text/javascript"> 
    /*
    This example is licensed under a 
    Creative Commons Attribution ShareAlike 4.0 International License.
    https://creativecommons.org/licenses/by-sa/4.0/
    
    Please note you have to mention 
    The Center of Mobile Learning with Digital Technology
    in the credits.
    */
    
    const BOARDID = 'board-0';

    var board = JXG.JSXGraph.initBoard(BOARDID, {
        boundingbox: [-2, 1.5, 2, -2.5],
        axis: true,
        showFullscreen: true,
    });
 </script> 
/*
This example is licensed under a 
Creative Commons Attribution ShareAlike 4.0 International License.
https://creativecommons.org/licenses/by-sa/4.0/

Please note you have to mention 
The Center of Mobile Learning with Digital Technology
in the credits.
*/

const BOARDID = 'your_div_id'; // Insert your id here!

var board = JXG.JSXGraph.initBoard(BOARDID, {
    boundingbox: [-2, 1.5, 2, -2.5],
    axis: true,
    showFullscreen: true,
});
Given are the contour lines of an objective function (red) and a constraint equation (blue). In order to find a maximum or minimum value of the objective function that simultaneously fulfills the constraint equation, the task is to find a point on the constraint curve fulfilling the necessary optimality condition. This is equivalent to find a point where the gradients of the objective function and of the constraint function are linearly dependent. That approach is known as the *method of Lagrange multipliers*.
Web references
Objective functions:

Constraints (function equals 0):
Equation:
0 =
0 =
Kontourlinien

    <div id="divexamples">
        <strong>Objective functions:</strong><br />
        <select id="examples" onchange="setText();" style="max-width: 600px; font-size: 80%">
            <option>x**2 - 6*x + y**2 - 4*y + 13</option>
            <option>x^2+y^2+x*y*2-2</option>
            <option>x^2 + y^2 + 2*x*y</option>
            <option selected="selected">sin(PI*(x**2 + y)) + cos(PI*y)</option>
            <option>1 / 9 * x ** 2 + y ** 2 - 1</option>
        </select><br />
        <strong>Constraints (function equals 0):</strong><br />
        <select id="examples2" onchange="setText2();" style="max-width: 600px; font-size: 80%">
            <option>-x^2-y+1</option>
            <option selected="selected">x^2+y-1</option>
            <option>-y+1</option>
            <option>y-0.1</option>
            <option>y-0.001</option>
            <option>4*(x-7)**2 + 4*(y-4)**2 - 5</option>
            <option>-x+y</option>
            <option>x + y - 2</option>
            <option>x^2 + y^2 + 2*x*y</option>
            <option>1 / 16 * x ** 2 + y ** 2 - 1</option>
            <option>x</option>
        </select>
    </div>
    <div id="divequation">
        <strong>Equation:</strong><br />

        0 =
        <input id="function" type="text" value="sin(PI*(x**2 + y)) + cos(PI*y)" style="font-size: 80%; width: 480px" />
        <button onClick="setObjective();plot();">select</button><br />
        0 =
        <input id="function2" type="text" value="x^2+y-1" style="font-size: 80%; width: 480px" />
        <button onClick="setConstraint();plot();">select</button><br />
        <strong>Kontourlinien</strong><br />
        <input id="niveauline" type="text" value="[-1.5,-0.5,0,0.5,1.5]" style="font-size: 80%; width: 480px" />
        <button onClick="setContour();plot();">set</button><br />

    </div>
// Define the id of your board in BOARDID

var board = JXG.JSXGraph.initBoard(BOARDID, {
    boundingbox: [-2, 1.5, 2, -2.5],
    axis: true,
    showFullscreen: true,
});

license

This example is licensed under a Creative Commons Attribution ShareAlike 4.0 International License.
Please note you have to mention The Center of Mobile Learning with Digital Technology in the credits.