K. Nicole Clark k-clark

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

**8.2.16 **– Suppose that at a point _{}in **R ^{3}**,
both first-order partial derivatives of a function

(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

mixed saddle point

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.