Chapter 5
Graphing and
Optimization
Section 1
First Derivative and Graphs

Objectives for Section 5.1
First Derivative and Graphs
■
Part 1:
■
Use the first derivative to determine when functions are increasing or decreasing.
■
Part II:
■
Use the first derivative to determine the local extrema of a function.
Barnett/Ziegler/Byleen Business Calculus 12e 2

Increasing/Decreasing Intervals
From a Graph f(x) is increasing on the interval (a, b) 𝑓 ′ 𝑓 ′ 𝑥 > 0 𝑥 = 0 on the interval (a, b) when x = b f(x) is decreasing on the interval (b, c) 𝑓 ′ 𝑥 < 0 on the interval (b, c)
Barnett/Ziegler/Byleen Business Calculus 12e 3

Derivatives on a Graph 𝑓 𝑓 ′
′ 𝑥 = 0: 𝑥 > 0: 𝑥 = 𝑐
1, 𝑐
2
, 𝑐
3
, 𝑐
5 𝑐
2
, 𝑐
3
∪ 𝑐
4
, 𝑐
5 𝑓 ′ 𝑥 < 0: −∞, 𝑐
1 𝑓 ′ 𝑥 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑:
∪ 𝑐
1
, 𝑐 𝑥 = 𝑐
2
4,
∪ (𝑐 𝑐
6
5
, 𝑐
6
∪ 𝑐
, 𝑐
7
3
)
, 𝑐
4
∪ 𝑐
6
, ∞
Barnett/Ziegler/Byleen Business Calculus 12e 4

Partition Points

Partition points are locations where there’s the potential for the derivative to change sign.

Partition points occur when the derivative is zero or undefined.
Barnett/Ziegler/Byleen Business Calculus 12e 5

Increasing/Decreasing Intervals
Using Calculus
1. Set 𝑓
′ 𝑥 = 0 𝑎𝑛𝑑 𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥
•
These x values are called partition points
2. Identify values of x that make 𝑓 ′ 𝑥 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
• These are also partition points
3. Plot partition points on a sign chart.
4. Plug in test numbers into 𝑓 ′ 𝑥 .
5. These will indicate intervals where 𝑓(𝑥) is increasing/decreasing.
6. Write your answer using interval notation.
Barnett/Ziegler/Byleen Business Calculus 12e 6

Example 1 𝑓 f ( x ) = x 2 + 6 x + 7 Find the intervals on which f(x) is increasing/decreasing.
′ 𝑥 = 2𝑥 + 6 𝑓 ′ 𝑥 𝑖𝑠 𝑛𝑒𝑣𝑒𝑟 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑.
0 = 2𝑥 + 6 𝑥 = −3 𝑓′(𝑥) − − −
0 𝑓(𝑥) 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 −3
−∞, −3
+ + + 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔
−3, ∞
Barnett/Ziegler/Byleen Business Calculus 12e 7

Example 2 f ( x ) = (1 – x ) 1/3 Find the intervals on which f(x) is increasing/decreasing.
𝑓 ′ 𝑥 =
−1
3 1−𝑥
−1
0 =
3 1 − 𝑥 2 3 𝑛𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑥 ≠ 1 𝑓′(𝑥) − − − 𝑁𝐷 − − −
1 𝑓(𝑥) 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔: −∞, 1 ∪ (1, ∞)
Barnett/Ziegler/Byleen Business Calculus 12e 8

Example 3 𝑓 𝑥 = 8 ln 𝑥 − 𝑥 2
Find the intervals on which f(x) is incr/decr. (Note: x > 0 because can’t take ln of a negative #) 𝑓′(𝑥) =
=
8 𝑥
8 𝑥−
− 2𝑥
2𝑥
2 𝑥
=
2(2−𝑥)(2+𝑥) 𝑥
=
=
8−2𝑥
2 𝑥
2(4−𝑥
2
) 𝑥
0 =
2(2−𝑥)(2+𝑥) 𝑥 𝑥 = 2, −2 𝑥 ≠ 0 𝑓′(𝑥) 𝑁𝐷 𝑓(𝑥)
0 2
𝐼𝑛𝑐𝑟: 0,2 𝐷𝑒𝑐𝑟: (2, ∞)
Barnett/Ziegler/Byleen Business Calculus 12e 9

Example 4 𝑥
2 𝑓 𝑥 = 𝑥+4
Find the intervals on which f(x) is incr/decr.
𝑓′ 𝑥 =
𝐿𝐻
′
−𝐻𝐿
′
𝐿 2 𝑥 + 4 2𝑥 − 𝑥 2 (1) 𝑓′(𝑥) = 𝑓 ′ 𝑥 = 𝑥 + 4 2 𝑥 2 + 8𝑥 𝑥 + 4 2
= 𝑥(𝑥 + 8) 𝑥 + 4 2 𝑥(𝑥 + 8)
0 = 𝑥 + 4 2
0 = 𝑥(𝑥 + 8 ) 𝑥 = 0, −8 x

-4 𝑓′(𝑥) + + 0 − − 𝑁𝐷 − − 0 + +
−8 −4 0 𝑓(𝑥) 𝐼𝑛𝑐𝑟: −∞, −8 ∪ 0, ∞ 𝐷𝑒𝑐𝑟: −8, −4 ∪ (−4,0)
Barnett/Ziegler/Byleen Business Calculus 12e 10

Example 5
Match each graph with it’s sign chart.
𝐴
𝐷
𝐵
𝐸
Barnett/Ziegler/Byleen Business Calculus 12e
𝐶
𝐹
11

Homework
Find increasing/decreasing intervals only!
Barnett/Ziegler/Byleen Business Calculus 12e 12

Objectives for Section 5.1
First Derivative and Graphs
■
Part 1:
■
Use the first derivative to determine when functions are increasing or decreasing.
■
Part II:
■
Use the first derivative to determine the local extrema of a function.
Barnett/Ziegler/Byleen Business Calculus 12e 13

Critical Values
In the first part of the lesson, you learned how to find partition points . These points were used to determine intervals for which the graph of f(x) is increasing/decreasing.
Critical values are the partition points where local extrema
(maxima/minima) might be located.
Barnett/Ziegler/Byleen Business Calculus 12e 14

Local Extrema
When the graph of a continuous function changes from rising to falling, a high point or local maximum occurs.
When the graph of a continuous function changes from falling to rising, a low point or local minimum occurs.
Theorem.
If f is continuous on the interval ( a , b ), c is a number in ( a , b), and f ( c ) is a local extremum, then c is a critical value.
Barnett/Ziegler/Byleen Business Calculus 12e 15

First Derivative Test
Let c be a critical value of f . That is, f ( c ) is defined, and either f

( c ) = 0 or f

( c ) is not defined. Construct a sign chart for f

( x ) close to and on either side of c .
f (x) left of c f (x) right of c f (c)
Decreasing
Increasing
Decreasing
Increasing
Increasing local minimum at c
Decreasing local maximum at c
Decreasing not an extremum
Increasing not an extremum
Barnett/Ziegler/Byleen Business Calculus 12e 16

Graphs & Local Extrema
 Let’s first look at some examples of graphs with local extrema…
Barnett/Ziegler/Byleen Business Calculus 12e 17

First Derivative Test f

( c ) = 0: Horizontal Tangent
Barnett/Ziegler/Byleen Business Calculus 12e 18

First Derivative Test f

( c ) = 0: Horizontal Tangent
Barnett/Ziegler/Byleen Business Calculus 12e 19

First Derivative Test f

( c ) is not defined but f ( c ) is defined
Barnett/Ziegler/Byleen Business Calculus 12e 20

First Derivative Test f

( c ) is not defined but f ( c ) is defined
Barnett/Ziegler/Byleen Business Calculus 12e 21

Finding Local Extrema
Using Calculus

Step 1: Find the partition points i.
Set 𝑓
′ ii.
𝑥 = 0 𝑎𝑛𝑑 𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥
Identify values of x that make 𝑓 ′ 𝑥 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 iii. Use the partition points to make a sign chart for 𝑓
′ 𝑥

Step 2: Identify critical values . (These are the partition points where f(x) has a value.)

Step 3: If there is a change in sign on the left and right of the critical point, then a local extrema exists.
Barnett/Ziegler/Byleen Business Calculus 12e 22

Example 1 𝑓 𝑥 = −2𝑥 3 + 3𝑥 2 + 120𝑥 Find the partition points, critical points, & local extrema.
𝑓 ′ 𝑥 = −6𝑥 2
0 = −6 𝑥 2
+ 6𝑥 + 120
− 𝑥 − 20
0 = −6 𝑥 − 5 𝑥 + 4 𝑥 = −4, 5 𝑓′(𝑥) − −
0
+ +
0
−4 5
Partition points: x = -4, 5 f(-4)=-304 f(5) =425
Critical values: x = -4, 5
− − f(-4)=-304 is a local min. f(5) =425 is a local max.
Barnett/Ziegler/Byleen Business Calculus 12e 23

Barnett/Ziegler/Byleen Business Calculus 12e

Example 2 𝑓 𝑥 =
6
Find the partition points, critical points, & local 𝑥+2 extrema.
𝑓 𝑥 = 6 𝑥 + 2 −1
−2 𝑓′(𝑥) = −6 𝑥 + 2
−6
0 = 𝑥 + 2 2 𝑥 = 𝑛𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑥 ≠ −2
Partition point: x = -2
Critical values: none, f(-2) = undefined
No extrema
Barnett/Ziegler/Byleen Business Calculus 12e 25

Barnett/Ziegler/Byleen Business Calculus 12e

Example 3 𝑓 𝑥 = 𝑥 3 − 6𝑥 2 + 9𝑥 + 1 Find the partition points, critical points, local extrema, & sketch the graph.
𝑓 ′ 𝑥 = 3𝑥 2
0 = 3 𝑥 2
− 12𝑥 + 9
− 4𝑥 + 3
0 = 3 𝑥 − 1 𝑥 − 3 𝑥 = 1, 3
Partition points: x = 1, 3 𝑓′(𝑥) + +
0
− −
0
+ + f(1)=5 f(3) =1
1 3
Critical values: x = 1, 3 f(1)=5 is a local max. f(3) =1 is a local min.
Barnett/Ziegler/Byleen Business Calculus 12e 27

Barnett/Ziegler/Byleen Business Calculus 12e

Example 4

Use the information below to sketch a graph of f. Assume f(x) is continuous on (-

,

).
y x
Barnett/Ziegler/Byleen Business Calculus 12e 29

Example 5

Use the information to sketch a graph of f. Assume f(x) is continuous on (-

,

).
y x
Barnett/Ziegler/Byleen Business Calculus 12e 30

Comparing the Graphs of 𝑓 and 𝑓′ 𝑓 𝑥 = 𝑥 3 − 6𝑥 2 + 9𝑥 + 1 𝑓 ′ 𝑥 = 3𝑥 2 − 12𝑥 + 9 f’(x) > 0: (
, 1) f’(x) = 0: x = 1 f ’(x) < 0: (1, 3) f’(x) = 0: x = 3 f’(x) > 0: (3, 
)
Barnett/Ziegler/Byleen Business Calculus 12e f’(x) > 0: (
, 1) f’(x) = 0: x = 1 f ’(x) < 0: (1, 3) f’(x) = 0: x = 3 f’(x) > 0: (3, 
)
31

Example 6

The graph of 𝑓 ′ is shown below. Match it with the correct graph of 𝑓 .
f’(x) < 0: (
, -2) f’(x) = 0: x = -2 f ’(x) > 0: (-2, 2) f’(x) = 0: x = 2 f’(x) < 0: (2, 
)
Barnett/Ziegler/Byleen Business Calculus 12e 32

Homework
Barnett/Ziegler/Byleen Business Calculus 12e 33