Graphical Illustration of Bivariate Constrained Optimization

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.