Lagrange Multipliers

The best point on a path

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• At a constrained maximum, the constraint curve is tangent to a contour of f.
• Lagrange's condition: $\nabla f = \lambda \nabla g$ together with $g = c$.
• The multiplier λ measures how sensitive the best value is to the constraint.

Steps

1. Fencing a field
2. The highest point on a path
3. Touch the contour
4. Lagrange’s condition
5. Watching the gradients line up
6. On a circle
7. Along the river