Consider a function 
in one dimension. If 
has a relative extremum at 
, then either 
or 
is not differentiable at
. Either the first or second derivative
tests may be used to locate relative extrema of the first kind.
A necessary condition for 
to have a minimum (maximum)
at 
is
 |
and
 |
A sufficient condition is 
and 
(
). Let 
, 
, ..., 
, but 
. Then 
has a local maximum at
if 
is odd and 
, and 
has a local minimum at
if 
is odd and 
. There is a saddle
point at 
if 
is even.
