Extrema of a function of two variables

The animation connects the Hessian of a two-variable function with bowls, inverted bowls and saddles. At a stationary point of a twice continuously differentiable function, a positive definite Hessian gives a strict local minimum, a negative definite Hessian a strict local maximum, and an indefinite Hessian a saddle. A semidefinite Hessian alone is inconclusive.

Before you watch

• Stationary points and gradients
• Second partial derivatives
• Quadratic forms and definite matrices