Legrange Multipliers

Legrange Multipliers allow us to optimize a function (maximize or minimize) subject to constraints (such as the solution must lie on a specific curve. Dan Klein has a nice tutorial on Lagrange Multipliers, including inequality constraints (less-than or greater-than). The MIT Math Department has a nice applet that shows the solution space for f(x,y) and g(x,y). Notice that as you move the point around, the gradient of f and g (shown as arrows) are parallel at solution points.