# Optimization constrained extremum problems

Introduction: In the previous section we optimized functions of two variables, with the variables unconstrained. However, it is also possible that these variables are constrained by a restriction.

Definition: A constrained extremum problem is given by
• Optimize $z(x,y)$                     (This is the object function)
• Subject to $g(x,y)=k$          (This is the restriction)
• Where $x \in D_1$, $,y \in D_2$      (This is the domain)
Example: An example of a constrained extremum problem is given below.
• Maximize $z(x,y)=2xy+3y$
• Subject to $4x+y=10$
• Where $x,y>0$

Remark: A constrained extremum problem is also called a constrained optimization problem..

In this section we discuss three methods to solve such a problem: Substitution method, First-order condition constrained extremum problem and First-order method Lagrange.

Required preknowlegde: Chapter 1: Functions of one variable, Chapter 2: Differentiation of functions of one variable, Chapter 3: Functions of two variablesChapter 4: Differentiation of functions of two variables, Section: Optimization functions of one variable, Section: Optimization functions of two variables.