# second derivative test

(1.7 hours to learn)

## Summary

In an optimization problem, a critical point (where the partial derivatives are zero) may be a local minimum or maximum, or a saddle point. The second derivative test is a way of testing optimality: a point is a (local) minimum if the Hessian matrix is positive definite.

## Context

This concept has the prerequisites:

- optimization problems
- higher-order partial derivatives (The test involves second derivatives.)
- positive definite matrices (If the matrix is positive definite, then the point is a local minimum.)

## Core resources (read/watch one of the following)

## -Paid-

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Section 5.3, "Quadratic forms and the second derivative test," pages 208-215

## Supplemental resources (the following are optional, but you may find them useful)

## -Free-

→ MIT Open Courseware: Multivariable Caclulus (2010)

Video lectures for MIT's introductory multivariable calculus class.

## -Paid-

→ Multivariable Calculus

An introductory multivariable calculus textbook.

Location:
Section 13.10, "Critical points of functions of two variables," pages 927-933

## See also

-No Additional Notes-