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.
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Increasing/Decreasing Intervals
From a Graph
f(x) is increasing on the interval (a, b)
𝑓 ′ 𝑥 > 0 on the interval (a, b)
𝑓 ′ 𝑥 = 0 when x = b
f(x) is decreasing on the interval (b, c)
𝑓 ′ 𝑥 < 0 on the interval (b, c)
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Derivatives on a Graph
𝑓 ′ 𝑥 = 0: 𝑥 = 𝑐1, 𝑐2 , 𝑐3 , 𝑐5
𝑓 ′ 𝑥 > 0:
𝑐2 , 𝑐3 ∪ 𝑐4 , 𝑐5 ∪ (𝑐5 , 𝑐6 )
𝑓 ′ 𝑥 < 0:
−∞, 𝑐1 ∪ 𝑐1 , 𝑐2 ∪ 𝑐3 , 𝑐4 ∪ 𝑐6 , ∞
𝑓 ′ 𝑥 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑:
Barnett/Ziegler/Byleen Business Calculus 12e
𝑥 = 𝑐4, 𝑐6 , 𝑐7
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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.
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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.
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Example 1
f (x) = x2 + 6x + 7 Find the intervals on which f(x) is
increasing/decreasing.
𝑓 ′ 𝑥 = 2𝑥 + 6
𝑓 ′ 𝑥 𝑖𝑠 𝑛𝑒𝑣𝑒𝑟 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑.
0 = 2𝑥 + 6
𝑥 = −3
𝑓′(𝑥)
− −−
0
𝑓(𝑥) 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 −3
−∞, −3
Barnett/Ziegler/Byleen Business Calculus 12e
+ + +
𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔
−3, ∞
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Example 2
f (x) = (1 – x)1/3 Find the intervals on which f(x) is
increasing/decreasing.
𝑓 ′ 𝑥 = −1
3
0=
1−𝑥 −2 3
−1
3 1−𝑥
2 3
𝑛𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛
𝑥≠1
𝑓′(𝑥)
− −−
𝑁𝐷
− −−
1
𝑓(𝑥) 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔: −∞, 1 ∪ (1, ∞)
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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) = 𝑥8 − 2𝑥
2(2−𝑥)(2+𝑥)
=
=
8 2𝑥 2
𝑥− 𝑥
𝑥
0 = 2(2−𝑥)(2+𝑥)
𝑥
2
= 8−2𝑥
𝑥
2)
= 2(4−𝑥
𝑥
𝑥 = 2, −2
𝑥≠0
𝑓′(𝑥) 𝑁𝐷 + + + 0 − − −
0
2
𝑓(𝑥) 𝐼𝑛𝑐𝑟: 0,2
𝐷𝑒𝑐𝑟: (2, ∞)
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Example 4
𝑓 𝑥 =
𝑥2
𝑥+4
Find the intervals on which f(x) is incr/decr.
𝑥(𝑥 + 8)
𝑥 2 + 8𝑥
=
𝑓′(𝑥) =
𝑓′ 𝑥 = 𝐿2
𝑥+4 2
𝑥+4 2
2
𝑥
+
4
2𝑥
−
𝑥
(1)
𝑥(𝑥 + 8)
′
𝑓 𝑥 =
0=
𝑥+4 2
𝑥+4 2
0 = 𝑥(𝑥 + 8)
𝑥 = 0, −8 x  -4
𝐿𝐻 ′ −𝐻𝐿′
𝑓′(𝑥) + +
0 − − 𝑁𝐷 − − 0
−8
𝑓(𝑥) 𝐼𝑛𝑐𝑟: −∞, −8 ∪ 0, ∞
Barnett/Ziegler/Byleen Business Calculus 12e
−4
++
0
𝐷𝑒𝑐𝑟: −8, −4 ∪ (−4,0)
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Example 5
Match each graph with it’s sign chart.
𝐴
𝐵
𝐶
𝐷
𝐸
𝐹
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Homework
Find increasing/decreasing
intervals only!
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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.
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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.
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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.
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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
Increasing
Decreasing
Decreasing
Decreasing
local minimum at
c
local maximum at
c
not an extremum
Increasing
Increasing
not an extremum
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Graphs & Local Extrema
 Let’s first look at some examples of graphs with local
extrema…
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First Derivative Test
f (c) = 0: Horizontal Tangent
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First Derivative Test
f (c) = 0: Horizontal Tangent
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First Derivative Test
f (c) is not defined but f (c) is defined
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First Derivative Test
f (c) is not defined but f (c) is defined
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Finding Local Extrema
Using Calculus
 Step 1: Find the partition points
i. Set 𝑓 ′ 𝑥 = 0 𝑎𝑛𝑑 𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥
ii. 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.
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Example 1
𝑓 𝑥 = −2𝑥 3 + 3𝑥 2 + 120𝑥 Find the partition points,
critical points, & local extrema.
𝑓 ′ 𝑥 = −6𝑥 2 + 6𝑥 + 120
0 = −6 𝑥 2 − 𝑥 − 20
0 = −6 𝑥 − 5 𝑥 + 4
𝑥 = −4, 5
𝑓′(𝑥)
−−
0
−4
+ +
0
−−
5
Partition points: x = -4, 5
f(-4)=-304
f(5) =425
Critical values: x = -4, 5
Barnett/Ziegler/Byleen Business Calculus 12e
f(-4)=-304 is a local min.
f(5) =425 is a local max.
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Barnett/Ziegler/Byleen Business Calculus 12e
Example 2
6
𝑥+2
𝑓 𝑥 =
Find the partition points, critical points, & local
extrema.
𝑓 𝑥 = 6 𝑥 + 2 −1
𝑓′(𝑥) = −6 𝑥 + 2 −2
−6
0=
𝑥+2 2
𝑥 = 𝑛𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛
𝑥 ≠ −2
Partition point: x = -2
Critical values: none, f(-2) = undefined
No extrema
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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 − 12𝑥 + 9
0 = 3 𝑥 2 − 4𝑥 + 3
0=3 𝑥−1 𝑥−3
𝑥 = 1, 3
Partition points: x = 1, 3
𝑓′(𝑥)
+ + 0 −− 0 + +
f(1)=5
1
3
f(3) =1
f(1)=5 is a local max.
Critical values: x = 1, 3
f(3) =1 is a local min.
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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
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Example 5
 Use the information to sketch a graph of f. Assume f(x) is
continuous on (-, ).
y
x
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Comparing the Graphs
of 𝑓 and 𝑓′
𝑓 𝑥 = 𝑥 3 − 6𝑥 2 + 9𝑥 + 1
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 = 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, )
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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, )
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Homework
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