Definition of local maximum and local minimum values
If for all in an open disk, then is a local maximum
If for all in an open disk, then is a local minimum
Definition of a critical point
A critical point is a point
in the domain of
such that
or one of the first partial derivatives does not exist
Second Derivatives Test and description of a saddle point
Let
be a critical point of
and let
If , then is a saddle point
If , then
if , then is a local minimum
if , then is a local maximum
If , then the Second Derivatives Test is inconclusive
Find the local maximum and minimum values and saddle point(s) of
the function. Then graph the surface using a window that shows these
characteristics.
Absolute extrema
If for all in the domain of , then is the absolute maximum
If for all in the domain of , then is the absolute minimum
Find the absolute maximum and minimum values of
on the set
.
;
is the closed triangular region with vertices
,
,
and
.