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 ![]() ![]() |
---|---|---|
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 ![]() ![]() |
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 ![]() ![]() |
Fact (1) | The Hessian matrix ![]() |
-- | Fact-Given combination direct |
2 | The restriction of ![]() ![]() ![]() |