Second derivative test for a function of two 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
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., .
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.