1. Home
  2. For business economics
  3. Chapter 5: Optimization
  4. Optimization functions of two variables
  5. First-order condition extremum

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.