The Hessian matrix of a function  is the matrix of partial second derivatives:

.

To get the entries in the Hessian matrix, find these derivatives as follows:

By symmetry,

The Hessian matrix is

.

The graph of  has a saddle point at  if and only

when evaluated at this point. 

Calculate the determinant of the Hessian at this point in terms of by subtracting the upper-right to lower-left product by from the upper-left to lower-right product; set this less than 0 and solve for .

Therefore, the graph of  has a saddle point at  if . The correct choice is therefore .

First, it must be established that the graph of  has a critical point at ; this holds if , so the first partial derivatives of  must be evaluated at :

Since , the graph of  does not have a critical point at . 

First, it must be established that the graph of  has a critical point at ; this holds if , so the first partial derivatives of  must be evaluated at :

The graph of  has a critical point at , so the Hessian matrix test applies.

The Hessian matrix  is the matrix of partial second derivatives

,

the determinant of which can be used to determine whether a critical point of  is a local maximum, a local minimum, or a saddle point. Find the partial second derivatives of :

 all are constant functions. 

,

so

The Hessian matrix, evaluated at , is

.

Its determinant is the upper-left to lower-right product minus the upper-right to lower-left product;

The determinant of the Hessian is negative, so the graph of  has a saddle point at .

First, it must be established that the graph of  has a critical point at ; this holds if , so the first partial derivatives of  must be evaluated at :

The graph of  has a critical point at , so the Hessian matrix test applies.

The Hessian matrix  is the matrix of partial second derivatives

,

the determinant of which can be used to determine whether a critical point of  is a local maximum, a local minimum, or a saddle point. Find the partial second derivatives of :

All four partial second derivatives are constant; the Hessian matrix at  is 

Its determinant is the upper-left to lower-right product minus the upper-right to lower-left product:

First, it must be established that the graph of  has a critical point at ; this holds if , so the first partial derivatives of  must be evaluated at :

The graph of  has a critical point at , so the Hessian matrix test applies.

The Hessian matrix  is the matrix of partial second derivatives

,

the determinant of which can be used to determine whether a critical point of  is a local maximum, a local minimum, or a saddle point. Find the partial second derivatives of :

The Hessian matrix at  is 

Its determinant is the upper-left to lower-right product minus the upper-right to lower-left product:

First, it must be established that the graph of  has a critical point at ; this holds if , so the first partial derivatives of  must be evaluated at :

The graph of  has a critical point at , so the Hessian matrix test applies.

The Hessian matrix  is the matrix of partial second derivatives

,

the determinant of which can be used to determine whether a critical point of  is a local maximum, a local minimum, or a saddle point. Find the partial second derivatives of :

The Hessian matrix at  is 

Its determinant is the upper-left to lower-right product minus the upper-right to lower-left product:

The determinant is positive, making  a local extremum. Since  is negative,  is a local maximum. 

First, it must be established that the graph of  has a critical point at ; this holds if , so the first partial derivatives of  must be evaluated at :

The graph of  has a critical point at , so the Hessian matrix test applies.

The Hessian matrix  is the matrix of partial second derivatives

,

the determinant of which can be used to determine whether a critical point of  is a local maximum, a local minimum, or a saddle point. Find the partial second derivatives of :

All four partial second derivatives are constants. The Hessian matrix at any point, including , is

;

Its determinant is the upper-left to lower-right product minus the upper-right to lower-left product:

Since the determinant of the Hessian is 0, the Hessian matrix test is inconclusive.

First, it must be established that  is a critical point of the graph of ; this holds if and only if both first partial derivatives are equal to 0 at this point. Find the partial derivatives and evaluate them at :

Thus, the graph of has a critical point at .

The Hessian matrix is the matrix of partial second derivatives

;

Find these derivatives and evaluate them at :

At , the Hessian matrix is

The determinant of this matrix is

Since the determinant of the Hessian matrix is positive, the graph of has a local extremum at ; since , a negative value, it is a local maximum.

First, it must be established that is a critical point of the graph of ; this holds if and only if both first partial derivatives are equal to 0 at this point. Find the partial derivatives and evaluate them at :

Thus, the graph of has a critical point at .

The Hessian matrix is the matrix of partial second derivatives

;

Find these derivatives and evaluate them at :

The Hessian matrix, evaluated at , ends up being the matrix . The determinant of the matrix is 0, which means that the Hessian matrix test is inconclusive.