Shows the geometry of using Lagrange multipliers to optimize a surface subject to one constraint. (corresponding notebook for users of Mathematica who want the commands used to generate these graphics)

Consider the function , called the "objective function", whose graph is the surface below.

Then add in the the circle , called the constraint curve.

In 3-space the constraint equation defines a cylinder. Intersecting this cylinder with the surface gives us a curve in 3-space.

Looking at this intersection curve in 3-space, it appears that the function attains two local maxima and two local minima along the constraint curve.

Another way to study the method of Lagrange multipliers is throuogh the objective function's level curves (i.e., the curves in the contour diagram). Here Mathematica uses shading to to indicate height (darker represents lower on the z-axis).

In our example, the constraint equation can also be represented in two dimensions.

So here we look at the relationship between the level curves of the surface and the constraint equation.

Trace with your finger along the constraint curve. Note how the objective function's values change as you do that. For example, traveling along the constraint from the point (2,0), in either direction, the objective function's values decrease (remember that darker shading represents values farther down the z-axis). So (2,0) yields a local maximum with respect to travel on the constraint curve.

Thus, it turns out that an extreme point of our 3D curve occurs when the constraint equation (circle) is tangent to a level curve. This observation leads us to note that the gradient vector of the objective function is parallel to the gradient of the constraint (since both are perpendicular to the tangent vector at the point of tangency between the constraint curve and the level curve).

This work is part of the Multivariable Calculus with Mathematica Project at the University of Oklahoma.