# First-order condition extremum

Introduction: A point $(x,y)$ such that $z'_x(x,y)=0$ and $z'_y(x,y)=0$ is called a stationary point of the function $z(x,y)$.

Theorem:
• An extremum location is either a stationary point or a boundary point.
• Not every stationary point is an extremum location.
• Not every boundary point is an extremum location.
Remark 1: Whenever a stationary point of a function is not an extremum location we call it a saddle point.

Remark 2: Contrary to functions of one variable (see First-order condition extremum) not every boundary point is an extremum location.