## Mathematics for Economists

## Chapter 10 : Optimization, several variables

### By Lund University

In the final chapter of this course we look at maximization and minimization of functions of several variables. Optimization can be either unconstrained or constrained. By constrained optimization, we mean that the variables must satisfy one or several constraints, we are not free to pick any values when we optimize the function. Unconstrained optimization is very similar to optimization of a function of a single variable. For constrained optimization, we will study a solution technique called the method of Lagrange.

## Optimization, several variables

In the first section of this final chapter we look at optimization of functions of several variables. In this section we only look at unconstrained optimization. We will look at necessary and sufficient conditions for a local optimum point and we will look at the extreme value theorem and optimization of convex and concave functions. The optimization problem is very similar to optimization of a function of one variable, replacing ordinary derivatives with partial derivatives and second order derivatives with the Hessian.

#### Optimizing a function of 2 variables, necessary conditions

#### Optimizing a function of 2 variables, sufficient conditions

#### Optimizing a function of 2 variables, additional results

#### Multivariable optimization, n variables

## Optimization: Problems

Exercises on multivariable optimization

#### Problem: Maximize function of two variables

#### Problem: Minimize function of two variables

#### Problem: Maximize profit function

#### Problem: Second derivative test

#### Problem: Maximize utility function

#### Problem: Maximize profit function

#### Problem: First- and second derivative test

#### Problem: Classify stationary points

#### Problem: Find extreme points

#### Problem: Find maximum and minimum points

#### Problem: Optimizing function of 2 variables

#### Problem: open, closed, bounded and compact sets

## Constrained optimization

In the final theory section of this course we will look at constrained optimization. Specifically, we will introduce something called the Lagrangian and the Lagrange multiplier and we will study the method of Lagrange. This method will help us to find the optimal point that at the same time satisfy our constraint.

#### Constrained optimization

## Constrained optimization: Problems

Exercises on constrained optimization