This Demonstration visualizes a classical example of constrained optimization using a Lagrange multiplier.

In the milkmaid problem [1], a milkmaid walks from her home at to the river at the point , draws a bucket of water, and brings it to the cow at . The problem seeks to minimize the total distance traveled, .

Use the slider to change the course of the river and drag the locators to change the positions of the maid and the cow. The red dashed arrows mark the path with minimal distance.

The curves of constant distance from to via a variable point are ellipses with and as focal points. At the point on the river where the distance is minimized, there must be a common tangent to the river and the ellipse of the minimal distance.