Critical points occur where both first order derivatives vanish:
So we have two critical points, (-1, 0) and (1, 0), both of which lie in the domain (on the boundary
). At these points, we get values of
and
.
If we focus on just the boundaries:
(1) If
, then
, and
, so there are no other critical points along this boundary.
(2) If we take
and
, we can parameterize the semicircular arc by considering
. So we can write
In the first case, we have either
or
. In the second, we can solve the quadratic to find
or
. At each critical point, we get
Over the domain
, we get an absolute maximum value of 30 at
, which corresponds to the point
.