Constrained extrema

This video visually demonstrates the geometric meaning of Lagrange multipliers in constrained optimization using a 3D animation. It uses the objective function f(x,y) = x^2 + y^2 and the constraint g(x,y) = x^2 + y^2/2 - 1 = 0 as examples. The animation shows that at the extremum points, where the level curves of the objective function are tangent to the constraint curve, their gradient vectors are parallel. This leads to the equation ∇f = λ∇g, which is then solved to find four specific critical points.

Before you watch

• Multivariable Calculus
• Partial Derivatives
• Gradient Vectors
• Level Curves / Contour Plots