Overview

Critical points occur where the gradient of a function is zero or undefined. They help identify maxima, minima, and saddle points in multivariable functions.

Gradient and Critical Points

The gradient of a multivariable function is a vector:

Finding Critical Points

To find the critical points, solve:

or where it is undefined.


Note

Ensure you check both where the gradient equals zero and where it is undefined, as these can also be critical points.

Second Derivative Test for Multivariable Functions

To classify critical points, we use the Hessian matrix, which is the matrix of second-order partial derivatives:

Eigenvalues of the Hessian

  1. If all eigenvalues are positive, the point is a local minimum.
  2. If all eigenvalues are negative, the point is a local maximum.
  3. If the eigenvalues have mixed signs, the point is a saddle point.

Tip

For functions of two variables $ the Hessian simplifies to:

With the discriminant :

  • and : local minimum.
  • and : local maximum.
  • : saddle point.
  • : test is inconclusive.

Lagrange Multipliers

To find critical points subject to a constraint use Lagrange multipliers:

where is a scalar (the Lagrange multiplier). Solve the system of equations formed by:

and the constraint .


Summary

  • Find critical points by setting .
  • Use the Hessian matrix and eigenvalues (or discriminant for two variables) to classify critical points.
  • Apply Lagrange multipliers for constrained optimization problems.