# Second derivative test for a function of multiple variables

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

## Contents

## Statement

Suppose is a function of a vector variable with coordinates , and suppose is a point in the domain of with coordinates . Suppose all the first-order partial derivatives of at equal zero.

Suppose that all the second-order partial derivatives of at (pure and mixed) exist and are continuous at and around . Note that Clairaut's theorem on equality of mixed partials thus applies and we get that . Also, continuous partials of a given order implies differentiable that many times tells us that is twice differentiable at . Combining all these pieces of information, we get that the Hessian matrix exists at and is a symmetric matrix.

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

The test is as follows. We begin by computing the Hessian matrix of at . This is a square matrix of real numbers. Further, due to Clairaut's theorem on equality of mixed partials, it is a symmetric matrix.

We can now state the test explicitly:

Condition on Hessian matrix | How would we check this condition? | Conclusion for at |
---|---|---|

The matrix is a positive definite matrix, i.e., the bilinear form induced by the matrix is positive definite. | Since the matrix is a symmetric matrix, it suffices to check that all the principal minors have positive determinant. | strict local minimum |

The matrix is a negative definite matrix, i.e., the bilinear form induced by the matrix is negative definite. | Check that the negative of the matrix is positive definite. Equivalently, the sign of the determinant of any principal minor is where is the order of matrices. | strict local maximum |

The matrix is a positive semidefinite matrix but not a positive definite matrix. | Since the matrix is a symmetric matrix, it suffices to check that all the principal minors have nonnegative determinant, but at least one of the determinants is zero. | inconclusive. However, if any of the determinants for odd order minors is strictly positive, we can rule out the possibility of a maximum. |

The matrix is a negative semidefinite matrix but not a negative definite matrix. | Check that the negative of the matrix is positive semidefinite but not positive definite. | inconclusive. However, if any of the determinants for odd order minors is strictly negative, we can rule out the possibility of a minimum. |

The matrix is neither positive semidefinite nor negative semidefinite. | Check that it satisfies none of the cases above. | saddle point |

## Relation with other tests

### Changing the number of variables

### Other tests to determine whether critical points give local extreme values

## Facts used

## Proof

### Case of positive definite matrix

**Given**: is a function of a vector variable with coordinates , and suppose is a point in the domain of with coordinates . Suppose all the first-order partial derivatives of at equal zero.

Suppose that all the second-order partial derivatives of at (pure and mixed) exist and are continuous at and around . Note that Clairaut's theorem on equality of mixed partials thus applies and we get that . This forces the Hessian matrix to be symmetric.

**Suppose further that the Hessian matrix is positive definite**.

**To prove**: has a strict local minimum at .

**Proof**: The proof needs to be completed to address a subtlety.

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | For any unit vector , the second-order pure directional derivative is positive. | Fact (1) | The Hessian matrix is positive definite | -- | Fact-Given combination direct |

2 | The restriction of to any straight line through has a strict local minimum at . Moreover, ... |