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 |
---|---|---|
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Saddle point | The Hessian matrix is neither positive semidefinite nor negative semidefinite. |
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Local minimum (reasoning similar to the single-variable second derivative test) | The Hessian matrix is positive definite. |
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Local maximum (reasoning similar to the single-variable second derivative test) | The Hessian matrix is negative definite. |
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Inconclusive, but we can rule out the possibility of being a local maximum. | The Hessian matrix is positive semidefinite but not positive definite. |
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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., ![]() |
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.