# Second derivative test for a function of two variables

From Calculus

This article describes an analogue for functions of multiple variables of the following term/fact/notion for functions of one variable: second derivative test

This article describes a test that can be used to determine whether a point in the domain of a function gives a point of local, endpoint, or absolute (global) maximum or minimum of the function, and/or to narrow down the possibilities for points where such maxima or minima occur.

View a complete list of such tests

## Statement

Suppose is a function of two variables . Suppose is a point in the domain of such that both the first-order partial derivatives at the point are zero, i.e., .

Suppose that all the second-order partial derivatives (pure and mixed) for exist and are continuous at and around . Note that by Clairaut's theorem on equality of mixed partials, this implies that .

The **second derivative test** helps us determine whether has a local maximum at , a local minimum at , or a saddle point at .

First, consider the Hessian determinant of at , which we define as:

Note that this is the determinant of the Hessian matrix:

We now have the following:

Case | Local maximum, local minimum, saddle point, or none of these? | Interpretation in terms of second derivative test for a function of multiple variables |
---|---|---|

Saddle point | The Hessian matrix is neither positive semidefinite nor negative semidefinite. | |

and (note that these together also force ) | Local minimum (reasoning similar to the single-variable second derivative test) | The Hessian matrix is positive definite. |

and (note that these together also force ) | Local maximum (reasoning similar to the single-variable second derivative test) | The Hessian matrix is negative definite. |

and one or both of and is positive (note that if one of them is positive, the other one is either positive or zero) | Inconclusive, but we can rule out the possibility of being a local maximum. | The Hessian matrix is positive semidefinite but not positive definite. |

and one or both of and is negative (note that if one of them is negative, the other one is either negative or zero) | Inconclusive, but we can rule out the possibility of being a local minimum | The Hessian matrix is negative semidefinite but not negative definite. |

All entries of the Hessian matrix are zero, i.e., are all zero | Inconclusive. No possibility can be ruled out. | The Hessian matrix is both positive semidefinite and negative semidefinite. Basically, we can't say anything. |

## Relation with other tests

### Changing the number of variables

- Second derivative test: The version for a function of one variable.
- Second derivative test for a function of multiple variables: The two-variable case is a special, and relatively tractable, subcase of the multiple-variable case.