View source for Second derivative test for a function of multiple variables

{{multivariable analogue of|second derivative test}}
{{maxmin test}}
==Statement==

Suppose <math>f</math> is a function of a vector variable <math>\vec{x}</math> with coordinates <math>x_1,x_2,\dots,x_n</math>, and suppose <math>\vec{c}</math> is a point in the domain of <math>f</math> with coordinates <math>c_1,c_2,\dots,c_n</math>. Suppose all the first-order [[partial derivative]]s of <math>f</math> at <math>\vec{c}</matH> equal zero. 

Suppose that all the second-order partial derivatives of <math>f</math> at <math>\vec{c}</math> (pure and [[second-order mixed partial derivative|mixed]]) exist and are continuous at and around <math>\vec{c}</math>. Note that [[Clairaut's theorem on equality of mixed partials]] thus applies and we get that <math>f_{x_ix_j}(\vec{c}) = f_{x_jx_i}(\vec{c})</math>. Also, [[continuous partials of a given order implies differentiable that many times]] tells us that <math>f</matH> is twice differentiable at <math>\vec{c}</math>. Combining all these pieces of information, we get that the [[Hessian matrix]] exists at <math>\vec{c}</math> and is a symmetric matrix.

The '''second derivative test''' helps us determine whether <math>f</math> has a local maximum at <math>\vec{c}</math>, local minimum at <math>\vec{c}</math>, or a [[saddle point]] at <math>\vec{c}</math>.

The test is as follows. We begin by computing the [[Hessian matrix]] of <math>f</math> at <math>\vec{c}</math>. This is a <math>n \times n</matH> 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:

{| class="sortable" border="1"
! Condition on Hessian matrix !! How would we check this condition? !! Conclusion for <math>f</math> at <math>\vec{c}</math>
|-
| 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 <math>(-1)^r</math> where <math>r</math> 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===

* [[Second derivative test]]
* [[Second derivative test for a function of two variables]]

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

* [[First derivative test for a function of multiple variables]]

==Facts used==

# [[uses::Hessian matrix defines bilinear form that outputs second-order directional derivatives]]

==Proof==

===Case of positive definite matrix===

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'''Given''': <math>f</math> is a function of a vector variable <math>\overline{x}</math> with coordinates <math>x_1,x_2,\dots,x_n</math>, and suppose <math>\overline{c}</math> is a point in the domain of <math>f</math> with coordinates <math>c_1,c_2,\dots,c_n</math>. Suppose all the first-order [[partial derivative]]s of <math>f</math> at <math>\overline{c}</matH> equal zero. 

Suppose that all the second-order partial derivatives of <math>f</math> at <math>\overline{c}</math> (pure and [[second-order mixed partial derivative|mixed]]) exist and are continuous at and around <math>\overline{c}</math>. Note that [[Clairaut's theorem on equality of mixed partials]] thus applies and we get that <math>f_{x_ix_j}(\overline{c}) = f_{x_jx_i}(\overline{c})</math>. This forces the Hessian matrix <math>H(f)(\overline{c})</math> to be symmetric.

'''Suppose further that the Hessian matrix <math>H(f)(\overline{c})</math> is positive definite'''.

'''To prove''': <math>f</math> has a strict local minimum at <math>\overline{c}</math>.

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

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || For any unit vector <math>\overline{u}</math>, the second-order pure directional derivative <math>D_{\overline{u}}(D_{\overline{u}}(f))(\overline{c})</math> is positive. || Fact (1) || The Hessian matrix <math>H(f)(\overline{c})</math> is positive definite || -- || Fact-Given combination direct
|-
| 2 || The restriction of <math>f</math> to any straight line through <math>\overline{c}</math> has a strict local minimum at <math>\overline{c}</math>. Moreover, ... || || ||
|}