Basic Calculus
Quarter 3 – Module 8:
Extreme Value Theorem
Basic Calculus
Alternative Delivery Mode
Quarter 3 – Module 8: Extreme Value Theorem
First Edition, 2020
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Basic Calculus
Quarter 3 – Module 8:
Extreme Value Theorem
Introductory Message
This Self-Learning Module (SLM) is prepared so that you, our dear learners,
can continue your studies and learn while at home. Activities, questions, directions,
exercises, and discussions are carefully stated for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide you step-bystep as you discover and understand the lesson prepared for you.
Pre-tests are provided to measure your prior knowledge on lessons in each
SLM. This will tell you if you need to proceed on completing this module or if you
need to ask your facilitator or your teacher’s assistance for better understanding of
the lesson. At the end of each module, you need to answer the post-test to self-check
your learning. Answer keys are provided for each activity and test. We trust that you
will be honest in using these.
In addition to the material in the main text, Notes to the Teacher are also
provided to our facilitators and parents for strategies and reminders on how they can
best help you on your home-based learning.
Please use this module with care. Do not put unnecessary marks on any part
of this SLM. Use a separate sheet of paper in answering the exercises and tests. And
read the instructions carefully before performing each task.
If you have any questions in using this SLM or any difficulty in answering the
tasks in this module, do not hesitate to consult your teacher or facilitator.
Thank you.
What I Need to Know
This module is designed and created to help you understand the Extreme Value
Theorem (EVT).
After reading this module, you will be able to:
a. illustrate the extreme value theorem. (STEM_BC11D-IIId-2)
1
What I Know
Sketch the graph of the given functions and find the minimum and maximum point
of these functions at a given interval. Copy the table on a sheet of paper and write
your answer on it.
Function
Graph
1. 𝑓(𝑥) = 2𝑥 − 5
interval: [0, 6]
2. 𝑓(𝑥) = 𝑥 3 + 3
interval: [−1, 2]
3. 𝑓(𝑥) = 𝑥 2
interval: [−5, 5]
4. 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 3𝑥 − 1
interval: [−1, 3]
5. 𝑓(𝑥) = 𝑥 4 − 2𝑥 2 + 1
interval: [−2, 2]
2
Minimum Point
and Maximum
Point
6. 𝑓(𝑥) = |𝑥 |
interval: [−2, 2]
7. 𝑓(𝑥) = 𝑥 3 − 1
interval: [−1, 2]
8. 𝑓(𝑥) = −|𝑥 |
interval: [−1, 1]
9. 𝑓(𝑥) = −𝑥 4 + 2𝑥 2 − 1,
interval: [−1, 1]
10. 𝑓(𝑥) = sin 𝑥
interval: [−𝜋, 𝜋]
3
Lesson
1
Extreme Value Theorem
Aren’t you wondering how the doctors or pharmacists figure out the dosage of the
medicine they give without causing any harm to the patient? Or how do the
companies calculate the minimum or maximum prices they charge for their products
to maximize their profit? This topic will give an understanding how they make it.
What’s In
Complete the table of values for the given function and sketch its graph. Write your
answer on a separate sheet of paper.
Given: 𝑓(𝑥) = 2𝑥 + 5
𝒙
-3
-2
-1
0
1
2
3
𝒇(𝒙)
1.
What can you say about the values of f(x)?
2.
Describe the graph of the function.
4
What’s New
Now let us focus on the graph of the function below, 𝑓(𝑥) = 2𝑥 + 5. Observe the points
from 𝑥 = −3 to 𝑥 = 3.
Figure 1
Have you noticed that the graph in Figure 1 of the given function is continuous and
increasing from left side to the right side of the Cartesian Plane? Now, take a closer
look at the graph of Figure 1 from 𝑥 = −3 to 𝑥 = 3.
a. What are the coordinates of the minimum point from 𝑥 = −3 to 𝑥 = 3?
b. What are the coordinates of the maximum point from 𝑥 = −3 to 𝑥 = 3?
5
What is It
Extreme Value Theorem (EVT)
This theorem states that a function 𝑓(𝑥) which is found to be continuous over a
closed interval [𝑎, 𝑏] is guaranteed to have extreme values in that interval.
An extreme value of 𝑓 or extremum, is either a minimum or maximum value of a
function.
•
•
A minimum value of 𝑓 occurs at some 𝑥 = 𝑐, if 𝑓(𝑐) ≤ 𝑓(𝑥) for all 𝑥 ≠ 𝑐 in that
interval.
A maximum value of 𝑓 occurs at some 𝑥 = 𝑐, if 𝑓(𝑐) ≥ 𝑓(𝑥) for all 𝑥 ≠ 𝑐 in that
interval.
Note: In this module, we limit our illustration of extrema (plural form of extremum)
to graphical examples. More detailed and computational examples will follow once
derivatives have been discussed.
Examples
1. Observe the graph of 𝑓(𝑥) = 2𝑥 + 5 in the interval [−3, 3] (Figure 1). Since the
maximum point is (3, 11) and its minimum point is (−3, −1), therefore, the
maximum value of the function is 11 and its minimum value is –1.
❖ Notice that the extremum is simply the y-coordinate of the maximum or
minimum point of the function.
2. Sketch the graph and find the minimum and maximum value of the function
𝑓(𝑥) = 5𝑥 2 + 2𝑥 − 3 at the interval [−3, 2].
Note: You may download DESMOS application or GEOGEBRA in your gadget to
graph
the
function.
You
may
also
use
this
link
https://www.desmos.com/calculator.
6
Solution:
Figure 2 shows the graph of the given
function using Desmos application.
Since
𝑓(𝑥) = 5𝑥 2 + 2𝑥 − 3
is
a
quadratic function, its graph is a
parabola which opens upward, so its
minimum point is its vertex at
(−0.2, −3.2) and the maximum point
in the interval [−3, 2] is (-3, 36).
Therefore, the minimum value is
−3.2 and the maximum value is 36.
Figure 2
3. Does the function 𝑓(𝑥) = cos 𝑥 at the interval (0, 2𝜋) have extrema (both maximum
and minimum value)? Explain your answer.
Solution:
Figure 3
Figure 3 shows that the given function is continuous, but the symbol used in
the given interval is a parenthesis “( )” which indicates that points
−𝜋
𝜋
, 0) and ( , 0) do not belong to the interval(0, 2𝜋). Take a look at the graph
2
2
−𝜋
𝜋
of the points ( , 0) and ( , 0) which uses hallow circle to indicate that they
2
2
(
are not part of the interval. Thus, the function 𝑓(𝑥) = cos 𝑥 has no absolute
maximum or minimum value because it is not within the closed interval.
7
4. Does the function 𝑓(𝑥) =
𝑥−1
at [−4, 4] have extrema? Explain your answer.
𝑥+1
Solution:
Figure 4
Observe the graph of the given function (Figure 4). Notice that the graph of the
function breaks since it will be undefined at 𝑥 = −1. Therefore, 𝑓(𝑥) =
𝑥−1
has
𝑥+1
no maximum or minimum value because it is not a continuous function.
8
What’s More
Identify the extrema (both minimum and maximum) of the given graphs below. If
there is no extrema, provide an explanation. Write your answers on a separate sheet
of paper.
1.
2.
3.
4.
5.
9
What I Have Learned
Express what you have learned in this lesson by answering the questions below.
Write your answers on a separate sheet of paper.
1. When can we say that a function or a graph has extrema (both minimum and
maximum value)?
2. How do we identify the minimum and maximum value of the graph of a function
at a given interval?
What I Can Do
Read and answer the word problem below. Write your answer on a separate sheet of
paper.
5𝑡
𝑡 +1
Suppose that 𝑐(𝑡) = 2
(in mg/ml) represents the concentration of a drug in a
patient’s bloodstream t hours after the drug was administered. Sketch the graph of
the drug given to the patient from 0 to 10 hours. How much is the maximum dosage
of drug that can be given to the patient? How much is the minimum amount of drug
that can be given to the patient?
10
Assessment
Determine if the given function will have extrema. If it has extrema, identify its
maximum and minimum value. If it has no extrema, provide an explanation. Use a
separate sheet of paper to answer the following.
Function
1. 𝑓(𝑥) = |𝑥 |
Interval
[0, 1]
2. 𝑓(𝑥) = |𝑥 |
3. 𝑓(𝑥) =
(0, 1)
1
[−2, 2]
𝑥
4. 𝑓(𝑥) = 𝑥 2 − 1
(−1, 2)
5. 𝑓(𝑥) = |𝑥 + 1|
[−2, 3]
6. 𝑓(𝑥) = |𝑥 + 1| + 3
[−2, 2]
7. 𝑓(𝑥) = 𝑥 4 − 2𝑥 2 + 1
[−1, 1]
8. 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 3𝑥 − 1
(−1, 1)
1
𝑥−1
[2, 4]
9. 𝑓(𝑥) =
10. 𝑓(𝑥) = sin 𝑥
[
−𝜋 𝜋
, ]
2 2
Additional Activity
Research five examples of the use of extreme value theorem in real life. Write your
answers on a separate sheet of paper.
Scoring Rubric: Two points will be given for every correct example.
11
Answer Key
What I know
12
13
What's In
14
What I Have Learned
What’s More
1. The maximum value is 3 and the
minimum value is 2.
2. The graph has no extrema because it
has no absolute maximum and
minimum value.
3. The graph has no extrema because it
is a discontinuous function.
4. The graph has no extrema because it
is a discontinuous function.
5. The maximum value is 2 and the
minimum value is 0.
What I Can Do
1. A function has extrema if it is a
continuous function over a closed
interval.
2. A minimum value of 𝑓 occurs at
some 𝑥 = 𝑐 if 𝑓(𝑐) ≤ 𝑓(𝑥) for all 𝑥 ≠
𝑐 in that interval. A maximum
value of 𝑓 occurs at some 𝑥 = 𝑐 if
𝑓(𝑐) ≥ 𝑓(𝑥) for all 𝑥 ≠ 𝑐 in that
interval.
Assessment
1. The minimum value is 0,
maximum value is 1.
2. The function has no extrema
because it is not over a closed
interval.
3. The function has no extrema
because it is a discontinuous
function.
4. The function has no extrema
because it is not over a closed
interval.
5. The minimum value is 0,
maximum value is 4.
6. The minimum value is 3,
maximum value is 6.
7. The minimum value is 0,
maximum value is 1.
8. The function has no extrema
because it is not over a closed
interval.
9. The function has no extrema
because it is a discontinuous
function.
10. The minimum value is -1,
maximum value is 1.
The maximum dosage of drug that can be
given to the patient is 2.5 mg/ml while the
minimum amount of dosage is 0.495
mg/ml.
Additional Activity
Answer may vary.
References
DepEd. 2013. Basic Calculus. Teachers Guide.
DepEd. 2013. General Mathematics. Teachers Guide.
15
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