This Demonstration shows the minimization or maximization of a function (blue) subject to a constraint (red). The green point shows the optimal solution.
The graphic on the left shows the level curves of and together with the gradients. On the right, a 3D graphic of the function is shown together with the constraint of projected onto the surface of .
For either the minimum or maximum, the gradients of the function and the constraint are parallel: , with the Lagrange multiplier . By moving the point around the plot region, you can see that this is a necessary condition for constrained optimization.