Graphical Illustration of Bivariate Constrained Optimization

This Demonstration illustrates graphically finding a constrained extrema of a function of two variables with a constraint . In this Demonstration . While the brown point moves on the unit circle, the cyan point moves on the curve, which is the intersection of the surface that is the graph of the function and the cylinder given by .

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To find the extrema of a function , subject to the constraint that , form an auxiliary function . The coordinates of the extrema satisfy three equations: , , and =0. The function is referred to as the Lagrangian, and as the Lagrange multiplier. One observes that the first of the three equations is simply , and that we really have three variables , so the system we solve is not overdetermined. This method assumes that the constraint equation defines a smooth (differentiable) closed curve, so that no points require special testing.
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