First-order condition extremum | Wiskunde op Tilburg University

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.