Second-order condition extremum

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
  • if $C(c,d)>0$ and $z_{xx}''(c,d)>0$, 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.
Here $C(x,y)$ is the criterion function, defined by

$$C(x,y)=z''_{xx}(x,y) \cdot z''_{yy}(x,y) - (z''_{xy}(x,y))^2.$$