OPTIMISATION PROBLEMS  
Analysis  
We are often only interested in either the maximum or minimum of a function. In order to solve these kinds of problem you should do the following:
Determine from which function you want to find out the maximum or minimum. The function could easily have more than one independent variable; if this is the case then find the relation between them so that you are left with just one unknown in the function.
Find the maximum or minimum you are looking for by applying the necessary conditions to their derivatives.
Evaluate the solution you obtain
Let's look at a couple of examples.
EXAMPLE 1 

A company has decided to improve its security by installing 9 alarms. A specialist points out to the company that given the kind of company that it is they can only opt for two types of alarms: type A or type B. Furthermore, he states that the company's security can be expressed as a tenth of the product of the number of type A alarms multiplied by the square of the number of type B alarms. How many of each type of alarm should the company have installed in order to maximise their security? 

a. Determine the
function
b. Find the maximum
c. Evaluate the solution


EXAMPLE 2 

There is a power station on the banks of a 1 km wide river. On the other side of the river there is a factory 8 km upstream. It costs €0.30/metre to run a cable on land and €0.50/metre to run it underwater. What is the cheapest way of running cable from the power station to the factory. 

a. Determine the function
b. Find the minimum
c. Evaluate the solutions

You can now do some exercises: EXERCISE 1 EXERCISE 2 EXERCISE 3
María José García Cebrian  
Spanish Ministry of Education. Year 2001  
Except where otherwise noted, this work is licensed under a Creative Common License