If is a (sufficiently differentiable) function of a single
variable and
has a relative minimum or maximum (generically
an extremum) at
then
.
Recall that a function may have without
being an
extremum.
If is a relative extremum of
, then
is a
relative extremum of
and
is a relative
extremum of
. So,
and
(In fact, this test applies to functions in any number of variables.)
As with functions of a single variable, there may be points
which are not relative extrema but for which
.
Recall that for a function of a single variable, one can look at the second derivative to test for concavity and thereby also the existence of a local minimum or maximum.
A (sufficiently smooth) function of one variable has a relative
extremum at
if
and
. If
and
, then
is a relative minimum and
if
and
, then
is a relative maximum.
Given a function of two variables we define a new function
If
then has a relative extremum at
(maximum if
and minimum if this second derivative is positive).
Conversely, if
then does not have a relative extremum at
.
When , this test yields no information.
Find the relative extrema of
.
The solutions to
are
and
. We compute the
and
.
Thus, the only potential relative extremum is at
.
We compute
. Thus,
is a relative maximum.
Find the extrema of
.
Setting both of these equal to zero, we find and
.
As
, the point
is not an extremum,
there are no local extrema of
.