**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 variables, Chapter 4: Differentiation of functions of two variables, Section: Optimization functions of one variable, Section: Optimization functions of two variables.