EXAMPLE 1 Find where the function f(x) = 3x* – 4x³ - 120x? + 1 is increasing and where it is decreasing. SOLUTION r'x) = 12x - 12x? - 240x = 12x(x - O)(*+O) To use the 1/D Test, we have to know where f'(x) > 0 and where f'(x) < 0. This depends on the signs of the three factors of f'(x), namely, 12x, x -O, and x +O. We divide the real line into intervals whose endpoints are the critical numbers O (smallest), 0 and ( (largest) and arrange our work in a chart. A plus sign indicates that the given expression is positive, and a negative sign indicates that it is negative. The last column of the chart gives the conclusion based on the 1/D Test. For instance, f'(x) < 0 for 0 < x < 5, so fis -Select- on (0, 5). (It would also be true to say that f is decreasing on the closed interval [0, 5]).) 12x x - 5 x + 4 r'(x) Interval x< -4 decreasing on (-*, -4) --Select--- V) on (-4, 0) decreasing on (0, 5) -4 5 Select--- ♥ on (5, )
EXAMPLE 1 Find where the function f(x) = 3x* – 4x³ - 120x? + 1 is increasing and where it is decreasing. SOLUTION r'x) = 12x - 12x? - 240x = 12x(x - O)(*+O) To use the 1/D Test, we have to know where f'(x) > 0 and where f'(x) < 0. This depends on the signs of the three factors of f'(x), namely, 12x, x -O, and x +O. We divide the real line into intervals whose endpoints are the critical numbers O (smallest), 0 and ( (largest) and arrange our work in a chart. A plus sign indicates that the given expression is positive, and a negative sign indicates that it is negative. The last column of the chart gives the conclusion based on the 1/D Test. For instance, f'(x) < 0 for 0 < x < 5, so fis -Select- on (0, 5). (It would also be true to say that f is decreasing on the closed interval [0, 5]).) 12x x - 5 x + 4 r'(x) Interval x< -4 decreasing on (-*, -4) --Select--- V) on (-4, 0) decreasing on (0, 5) -4 5 Select--- ♥ on (5, )
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter5: A Survey Of Other Common Functions
Section5.4: Combining And Decomposing Functions
Problem 14E: Decay of Litter Litter such as leaves falls to the forest floor, where the action of insects and...
Related questions
Question
![y
EXAMPLE 1
Find where the function f(x) = 3x - 4x3 – 120x² + 1 is increasing and where it is decreasing.
500-
f'(x) = 12x3 – 12x2 – 240x = 12x x -
x-O)(x+O)
X
SOLUTION
-4
-2
2
4
6
–500
To use the I/D Test, we have to know where f'(x) > 0 and where f'(x) < 0. This depends on the signs of the three factors of f'(x), namely,
. We divide the real line into intervals whose endpoints are the critical numbers
12х, х —
and x +
(smallest), 0 and
-1000
(largest) and arrange our work in a chart. A plus sign indicates that the given expression is positive, and a negative sign indicates that it is
negative. The last column of the chart gives the conclusion based on the I/D Test. For instance, f'(x) < 0 for 0 < x < 5, so f is ---Select--- ♥
on (0, 5). (It would also be true to say that f is decreasing on the closed interval [0, 5].)
-1500
Video Example
Interval
12x
х — 5
x + 4
f'(x)
X < -4
decreasing on (-0, -4)
-4 < x < 0
+
---Select--- ♥] on (-4, 0)
0 < x < 5
decreasing on (0, 5)
x > 5
---Select--- V
on (5, 0)
+
The graph of f shown in the figure confirms the information in the chart.
+
+](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8796f270-f66d-4e05-adbc-919a3794eb0d%2Fbc5342df-8b3d-4dae-ba64-823b8ca7a8e4%2Fqdwk2w7_processed.png&w=3840&q=75)
Transcribed Image Text:y
EXAMPLE 1
Find where the function f(x) = 3x - 4x3 – 120x² + 1 is increasing and where it is decreasing.
500-
f'(x) = 12x3 – 12x2 – 240x = 12x x -
x-O)(x+O)
X
SOLUTION
-4
-2
2
4
6
–500
To use the I/D Test, we have to know where f'(x) > 0 and where f'(x) < 0. This depends on the signs of the three factors of f'(x), namely,
. We divide the real line into intervals whose endpoints are the critical numbers
12х, х —
and x +
(smallest), 0 and
-1000
(largest) and arrange our work in a chart. A plus sign indicates that the given expression is positive, and a negative sign indicates that it is
negative. The last column of the chart gives the conclusion based on the I/D Test. For instance, f'(x) < 0 for 0 < x < 5, so f is ---Select--- ♥
on (0, 5). (It would also be true to say that f is decreasing on the closed interval [0, 5].)
-1500
Video Example
Interval
12x
х — 5
x + 4
f'(x)
X < -4
decreasing on (-0, -4)
-4 < x < 0
+
---Select--- ♥] on (-4, 0)
0 < x < 5
decreasing on (0, 5)
x > 5
---Select--- V
on (5, 0)
+
The graph of f shown in the figure confirms the information in the chart.
+
+
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