The second partial derivative test classifies a critical point 
of a twice continuously differentiable function 
.
Define
 |
If 
and 
,
the point is a local minimum; if 
and 
, it is a local maximum.
If 
,
the point is a saddle point. When 
, the test is inconclusive. These cases are the two-dimensional
form of classifying the Hessian by its definiteness.
