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.
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