CK12 Calculus - Single Variable

Optimization

LearningObjectives
A studentwill be able to:

• Use the First and SecondDerivativeTests to find absolutemaximumand minimumvaluesof a function.


• Use the First and SecondDerivativeTests to solveoptimizationapplications.

Introduction
In this lessonwe wish to extendour discussionof extremaand look at the absolutemaximumand minimum
valuesof functions.We will then solvesomeapplicationsusingthesemethodsto maximizeand minimize
functions.
AbsoluteMaximumand Minimum
We beginwith an observationaboutfindingabsolutemaximumand minimumvaluesof functionsthat are
continuouson a closedinterval.Supposethat is continuouson a closedinterval Recallthat we
can find relativeminimaand maximaby identifyingthe criticalnumbersof in and then applying
the SecondDerivativeTest. The absolutemaximumand minimummustcomefrom eitherthe relativeextrema
of in or the valueof the functionat the endpoints, or Hencethe absolutemaximum
or minimumvaluesof a function that is continuouson a closedinterval can be foundas follows:

1. Find the valuesof for eachcriticalvaluein ;


2. Find the valuesof the function at the endpointsof ;


3. The absolutemaximumwill be the largestvalueof the numbersfoundin 1 and 2; the absoluteminimum
   will be the smallestnumber.
   The optimizationproblemswe will solvewill involvea processof maximizingand minimizingfunctions.Since
   mostproblemswill involvereal applicationsthat one findsin everydaylife, we needto discusshow the
   propertiesof everydayapplicationswill affect the moretheoreticalmethodswe havedevelopedin our
   analysis.Let’s start with the followingexample.
   Example1:
   A companymakeshigh-qualitybicycletires for both recreationaland racingriders.The numberof tires that
   the company sells is a function of the price charged and can be modeled by the formula
   where is the pricedchargedfor eachtire in dollars.At what price
   is the maximumnumberof tires sold?How manytires will be sold at that maximumprice?
   Solution:
   Let’s first look at a graphand makesomeobservations.Set the viewingwindowrangeson your graphing
   calculatorto for and for The graphshouldappearas follows: