A point is called a critical point of a continuously differentiable function if
If the second partial derivatives of are continuous at as well,
the type of the critical point can be classified in terms of the eigenvalues of the
A critical point is called
if all eigenvalues are different from zero and have the same sign;
hyperbolic or a saddle point,
if there exist eigenvalues with different signs;
if at least one eigenvalue is zero and all nonzero eigenvalues have the same sign.
At an elliptic point, has a local extremum.
This might also be the case for a parabolic point.
However, a decision requires more information, since higher order partial derivatives can influence the local behavior of .
At a saddle point, there exist directions with increasing and decreasing function values.