a.

Critical points occur where
. The exponential factor is always positive, so we have

b. As the previous answer established, the critical point occurs at (-3, 8) if
and
.
c. Check the determinant of the Hessian matrix of
:

The second-order derivatives are




so that the determinant of the Hessian is


The sign of the determinant is unchanged by the exponential term so we can ignore it. For
and
, the remaining factor in the determinant has a value of 4, which is positive. At this point we also have

which is negative, and this indicates that (-3, 8) is a local maximum.