ABSOLUTE MINIMUM AND MAXIMUMS By: Hannah Ahluwalia and Anita Vellaichamy

ABSOLUTE MINIMUM AND MAXIMUMS

By: Hannah Ahluwalia and Anita Vellaichamy

LESSON OBJECTIVES

In this lesson, you will learn how to find

critical numbers analytically and

graphically, as well as using them to

determine the absolute minimum and

maximum values of a function

INTRODUCTION

In order to continue with the lesson,

you must know how to find the

derivative of a function

EXAMPLE

Find the derivative of a

function through the

power rule

INTRODUCTION PART 2

You must also know how to solve for

the zeroes of a function. This is done

simply by setting an equation equal to

zero and solving for x.

EXAMPLE 2

CRITICAL NUMBERS

Now that you know these basics, we

may move into the first part of the

lesson: finding critical numbers

CRITICAL NUMBERS CONT.

A critical point of a function is where

or

CRITICAL POINTS CONT.

To find the critical points of a function

analytically, you must take the

derivative of the function. Once you

have the derivative, set it equal to zero

and solve for X

CRITICAL POINTS EX. 1

Points A and B are

critical points because A

B

CRITICAL POINT EX. 2

These are your critical numbers

NOW TRY ONE FOR YOURSELF!

Find the critical points

of the function.

CRITICAL POINT QUIZ

A.

B. ,

C.

D.

LESSON CONT.

Now that you know how to find

critical points let’s learn how to you

them! If you want to review critical

points, feel free to go back at any time. Review Continue

LESSON CONT.

You must plug each x-value into the

function. This includes critical points

that you solved for, and the endpoints

of the function.

Continue

Back

ABS. MAXIMUM AND MINIMUM EX.1

Function:

1. Get the derivative (you can use the power rule here):

2. Factor the derivative you found:

3. Set each equal to zero to find the critical points:

This function has two critical points.

ContinueBack

http://tutorial.math.lamar.edu/Classes/CalcI/AbsExtrema.aspx

EXAMPLE CONT.

Next, lay out what you need to solve, and evaluate.

critical points

endpoints

Abs. minimumAbs. maximum

Continue

Back

FINISHING THE EXAMPLE

Now state your conclusion, depending on what the

question asks you, you may need to say it one of two

ways:

1. The abs. maximum occurs at and the abs. minimum

occurs at

2. The abs. maximum value is , and the abs. minimum

value is Back Continue

NOW YOU TRY ONE!

Start by finding the critical points

of:

Review how to find these Continue

to answer

https://mathway.com/examples/Calculus/Applications-of-Differentiation/Finding-the-Absolute-Maximum-and-Minimum-on-the-Given-Interval?id=825

DID YOU GET IT?

𝑥=0,2If no, try againIf yes, continue

EXAMPLE CONT.

Now plug the x- values into the

equation and find the absolute

maximum and minimum.

Back Continue

EXAMPLE CONT.

Did you get the abs. maximum at

with a value of and an abs.

minimum at and a value of ?

If no, try again

If yes, continue

LESSON CONT.

If you are finding the abs. maximum

and minimum by using your calculator,

you will need to graph the function to

find the critical points. Back to example problem

Continue with lesson

CALCULATOR EX 1.

Function: ,

Start again by finding the derivative:

Next, graph the derivative function, to see where it crosses

the x- axis ( these x- values are the critical points). Make

sure to use the interval they gave you.

BackContinue to example

http://tutorial.math.lamar.edu/Classes/CalcI/AbsExtrema.aspx

EXAMPLE CONT.

To help identify critical points on the graph, you can

use the following calculator directions: 2nd ,

calculate, zero, click left bound, right bound, and

one more time for the guess.

You should find these as the critical values:

Back

Continue

EXAMPLE CONT.

Next, test the values into the equation, include critical points and

endpoints.

𝑓(0)=100.0

𝑓(0.604)=102.4756

𝑓(0.9661)=102.2368

𝑓(2.1755)=107.1880

𝑓(2.5369)=106.9492

𝑓( 3.7463)=111.9004

𝑓(4)=111.7121

Abs. minimum

Abs. maximum

Back Finish the example

FINISH THE EXAMPLE

To finish identify the Abs. minimum and Abs.

maximum.

Abs. maximum occurs at , with a value of

Abs. minimum occurs at , with a value of

Try one on your own

Back

NOW YOU TRY ONE!

First find the derivative of

Review how to find derivative

Continue to answer

http://archives.math.utk.edu/visual.calculus/3/max.1/3.html

PROBLEM CONT.

Did you get this as the derivative?

If no, try again If yes, continue

PROBLEM CONT.

Did you get these as the critical points?

If no, try again

If yes, continue

PROBLEM CONT.

Were these your final answers?

Abs. maximum at with a value of

Abs. minimum at with a value of

If yes, continue

If no, try again

GOOD JOB!

If you want to review, go back, if

you would like to move forward to

the quiz, click continue. Review from beginning of lesson

Continue to quiz

QUESTION 1

What is the critical points of 𝑓 (𝑥 )=𝑥2−5 𝑥+7 , [−1,3 ]

2.5 8.4 4.2

Review topic http://archives.math.utk.edu/visual.calculus/3/max.1/1.html

QUESTION 2

Find the Abs. maximum value of

**Use calculator for this question http://archives.math.utk.edu/visual.calculus/3/max.1/4.html

Review

QUESTION 3

Find the Abs. minimum value of

http://archives.math.utk.edu/visual.calculus/3/max.1/2.html

Review

QUESTION 4

Now find both extrema of

**Use calculator for this problemContinue to first part of question

http://archives.math.utk.edu/visual.calculus/3/max.1/7.html

QUESTION 4 CONT.

Abs. maximum is at

14

𝑥=?

Review

QUESTION 4 CONT.

Abs. minimum is at

Review

GREAT JOB!

Now you can find both absolute

extrema, but feel free to review if you

need to.

Back to beginning Finis

h

INCORRECT!

Try this one again!

INCORRECT!

Try this one again!

Back to quiz question 1

CORRECT!

After taking the derivative of the function, you

should get

Then, set this equal to zero.

And solve!

You should get ,

CORRECT!

Next questionBack to question 1

CRITICAL NUMBERS CONT.

A critical point of a function is where

or

Back to quiz question 1

CRITICAL NUMBERS CONT.

A critical point of a function is where

or

Back to lesson

EXAMPLE

Find the derivative of a

function through the

power rule

Back to example

CRITICAL POINT EX. 2

These are your critical numbers

Back to example

ABS. MAXIMUM AND MINIMUM EX.1

Function:

1. Get the derivative (you can use the power rule here):

2. Factor the derivative you found:

3. Set each equal to zero to find the critical points:

This function has two critical points.

Back

INCORRECT!

Try this one again!

Back to quiz question 2

CORRECT!

Abs. Maximum

Continue to question 3Back to question 2

EXAMPLE CONT.

Next, lay out what you need to solve, and evaluate.

critical points

endpoints

Abs. minimumAbs. maximum

Back to question 2

EXAMPLE CONT.

Next, lay out what you need to solve, and evaluate.

critical points

endpoints

Abs. minimumAbs. maximum

Back to question 3

INCORRECT!

Try this one again!

Back to quiz question 3

CORRECT!

Abs. minimum value

Continue to question 4

Back to question 3

EXAMPLE CONT.

Next, lay out what you need to solve, and evaluate.

critical points

endpoints

Abs. minimumAbs. maximum

Back to question 4 part 1

INCORRECT!

Try this one again!

Back to quiz question 4 part 1

CORRECT!

Abs. Maximum

Continue to next part of question 4

Back to part 1

INCORRECT!

Try this one again!

Back to quiz question 4 part 2

CORRECT!

Abs. Minimum

Finish quiz

Back to part 2

EXAMPLE CONT.

Next, lay out what you need to solve, and evaluate.

critical points

endpoints

Abs. minimumAbs. maximum

Back to question 4 part 2

FINISH!