Second derivative test for a function of multiple variables

Statement

Suppose $f$ is a function of a vector variable ${\overline {x}}$ with coordinates $x_{1},x_{2},\dots ,x_{n}$, and suppose ${\overline {c}}$ is a point in the domain of $f$ with coordinates $c_{1},c_{2},\dots ,c_{n}$. Suppose all the first-order partial derivatives of $f$ at ${\overline {c}}$ equal zero.

Suppose that all the second-order partial derivatives of $f$ at ${\overline {c}}$ (pure and mixed) exist and are continuous at and around ${\overline {c}}$. Note that Clairaut's theorem on equality of mixed partials thus applies and we get that $f_{x_{i}x_{j}}({\overline {c}})=f_{x_{j}x_{i}}({\overline {c}})$.

The second derivative test helps us determine whether $f$ has a local maximum at ${\overline {c}}$, local minimum at ${\overline {c}}$, or a saddle point at ${\overline {c}}$.

The test is as follows. We begin by computing the Hessian matrix of $f$ at ${\overline {c}}$. This is a $n\times n$ 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 $f$ at ${\overline {c}}$
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 determinant of any principal minor is $(-1)^{r}$ where $r$ 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.	Indeterminate.
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.	Indeterminate.
The matrix is neither positive semidefinite nor negative semidefinite.	Check that it satisfies none of the cases above.	saddle point

Relation with other tests

• Second derivative test for a function of two variables