K. Nicole Clark                                                                                           k-clark

4/30/2004

URL: people.tamu.edu/~knc8928/8.2.16.pdf

8.2.16 – Suppose that at a point in R3, both first-order partial derivatives of a function are zero, and the matrix of second-order partial derivatives is one of those given below.  In each case, tell whether is the location of a maximum, a minimum, or a saddle point.

(a)

(b)

Solution

We will use the chart found on page 426 of Linearity as a guide in this problem.

signs                                              behavior

all positive                                        minimum

all negative                                       maximum

some zero, others all one sign          more information needed

(a)    We will use the rule , where A is the matrix given above, to find the eigenvalues.

This simplifies to , which has roots

Since the roots have mixed signs, from the chart above we know that is a saddle point.

(b)   We will use the same method we used in (a) to find the eigenvalues in this case.

This simplifies to , which has roots

Once again, the eigenvalues have mixed signs, thus is again a saddle point.