Introduction: The Second-order partial derivatives can be used to determine whether a Stationary point is a minimum or a maximum.
Definition: For a stationary point (c,d) of the function z(x,y), that is not on the boundary of the domain, it holds that
C(x,y)=z''_{xx}(x,y) \cdot z''_{yy}(x,y) - (z''_{xy}(x,y))^2.
Definition: For a stationary point (c,d) of the function z(x,y), that is not on the boundary of the domain, it holds that
- if C(c,d)>0 and zxx″, then z(c,d) is a minimum,
- if C(c,d)>0 and z_{xx}''(c,d)<0, then z(c,d) is a maximum,
- if C(c,d)<0, then (c,d) is a saddle point.
C(x,y)=z''_{xx}(x,y) \cdot z''_{yy}(x,y) - (z''_{xy}(x,y))^2.