Suppose 
is a function of 
that is twice differentiable
at a stationary point 
.
1. If 
,
then 
has a local minimum at 
.
2. If 
,
then 
has a local maximum at 
.
The extremum test gives slightly more general conditions under which a function with 
is a maximum or minimum.
If 
is a two-dimensional function that has a local
extremum at a point 
and has continuous partial derivatives at this point, then 
and 
. The second partial
derivatives test classifies the point as a local
maximum or local minimum.
Define the second derivative test discriminant as
 |  |  |
(1)
|
 |  |  |
(2)
|
Then
1. If 
and 
,
the point is a local minimum.
2. If 
and 
,
the point is a local maximum.
3. If 
,
the point is a saddle point.
4. If 
,
higher order tests must be used.
