Transcript 3 1 Extrema Interval

3-1: Extrema on an Interval
Objectives:
Assignment:
1. To define and distinguish
between absolute
• P. 169: 1, 9, 11
extrema and relative
extrema
2. To find extrema on a
closed interval
• P. 169-171: 13-18, 19-27
odd, 33, 39, 41, 55-58,
61, 63-66
Warm-Up
Draw a continuous function connecting two
endpoints.
Absolute
Maximum
Relative
Maximum
Absolute
Minimum
Relative
Minimum
How many absolute
minimum points do
you have? How
many absolute max
points do you have?
Where are they?
Warm-Up
Draw a continuous function connecting two
endpoints.
Not a
Absolute
Maximum
What if your curve is
not continuous?
Objective 1
You will be able to define and distinguish
between absolute extrema and relative
extrema
Extrema
Let 𝑓 be defined on an interval 𝐼 containing 𝑐.
𝑓 𝑐 is the maximum of
𝑓 on 𝐼 if 𝑓(𝑐) ≥ 𝑓(𝑥) for
all 𝑥 in 𝐼
𝑓 𝑐 is the minimum of
𝑓 on 𝐼 if 𝑓(𝑐) ≤ 𝑓(𝑥) for
all 𝑥 in 𝐼
The minimum or maximum of a function are
the extreme values, or extrema, of the
function.
Extreme Value Theorem
If 𝑓 is continuous on a closed interval 𝑎, 𝑏 , then 𝑓 has
both a maximum and a minimum on the interval.
Is it possible to
have more than
one minimum or
maximum?
Objective 2
You will be able to find
extrema on a closed interval
Relative Extrema
If there is an open
interval containing 𝑐 on
which 𝑓 𝑐 is a
maximum, then 𝑓 𝑐 is a
relative maximum of 𝑓.
𝑓 has a relative
maximum at 𝑐, 𝑓 𝑐 .
𝑓 has a relative
minimum at 𝑐, 𝑓 𝑐 .
If there is an open
interval containing 𝑐 on
which 𝑓 𝑐 is a
minimum, then 𝑓 𝑐 is a
relative minimum of 𝑓.
Exercise 1
Find the value of the derivative at each indicated value.
1.
𝑦=
9 𝑥 2 −3
𝑥3
, 3, 2
2.
𝑦 = 𝑥 , 0, 0
2.
𝑦 = sin 𝑥,
3𝜋
, −1
2
𝜋
,1
2
,
These are
called critical
values
Critical Numbers
Let 𝑓 be defined at 𝑐. If 𝑓′(𝑐) = 0 or if 𝑓 is
not differentiable at 𝑐, then 𝑐 is a critical
number of 𝑓.
Relative Extrema
Relative Extrema Occur Only at Critical
Numbers
If 𝑓 has a relative minimum or relative maximum at
𝑥 = 𝑐, then 𝑐 is a critical number of 𝑓.
Where 𝑓′ 𝑥 = 0
Where 𝑓′ 𝑥 is undefined
Exercise 2
Verify that 𝑥 = 0 is a critical number of
3
𝑓 𝑥 = 𝑥. Does 𝑓(𝑥) have a relative
minimum or maximum at 𝑥 = 0?
Neither minimum nor maximum
Exercise 3
Write the converse and contrapositive of the
statement below. Then determine the truth
value of each statement.
If f has a relative minimum or relative maximum
at 𝑥 = 𝑐, then 𝑐 is a critical number of 𝑓.
If 𝑥 = 𝑐 is a critical
number of 𝑓(𝑥), then
𝑓(𝑐) is a potential
minimum or
maximum.
Candidate
Closed Interval Test
Candidates Test
To find the extrema of a continuous function 𝑓
on a closed interval 𝑎, 𝑏 :
Find the
critical
Step 1
numbers
of 𝑓 in
(𝑎, 𝑏).
Evaluate
𝑓 at each
Step 2
critical
number in
(𝑎, 𝑏).
Evaluate 𝑓
at each
Step 3
endpoint of
𝑎, 𝑏 .
The least of
these is the
minimum;
Step 4
the greatest
is the
maximum.
Exercise 4
Find the extrema of
𝑓(𝑥) = 3𝑥 4 − 4𝑥 3 on the interval
−1, 2 .
Exercise 5
Find the extrema of 𝑓(𝑥) = 2𝑥 − 3𝑥 2/3 on the
interval −1, 3 .
Exercise 6
Find the extrema of 𝑓(𝑥) = 2 sin 𝑥 − cos2𝑥 on
the interval 0, 2𝜋 .
3-1: Extrema on an Interval
Objectives:
Assignment:
1. To define and distinguish
between absolute
• P. 169: 1, 9, 11
extrema and relative
extrema
2. To find extrema on a
closed interval
• P. 169-171: 13-18, 19-27
odd, 33, 39, 41, 55-58,
61, 63-66