The function
has critical points where the first partial derivatives simultaneously vanish:
The point (3, 3) lies within the disk
. The Hessian for
is
which is positive definite for all
. By the second partial derivative test, this indicates the critical point (3, 3) is the site of a minimum, and we have
.
Meanwhile, when we consider the boundary, we can set
and
, so that
can be expressed as a function of one variable
:
Then the critical points of
occur at
where
is any integer. We restrict
to omit redundancies, leaving us with just
and
. We then have
which indicates that a minimum occurs at
, and a maximum at
. At these points we have values of
and
.
So the absolute minimum of
over the disk is -18, and the absolute maximum of
over the disk is
.