Consider the equation below. f(x) = 6 sin(x) + 6 cos(x), 0 sxs 2n Exercise (a) Find the interval on which f is increasing. Find the interval on which f is decreasing. Step 1 For f(x) = 6 sin(x) + 6 cos(x), we have f'(x) = 6 cos (x) – 6 sin (x 6 cos (x) – 6 sin(x) If this equals 0, then we have cos(x) = sin (x) sin(x) which becomes tan(x) = Hence, in the interval 0 s x s 27, f'(x) = 0 when x = or 57 X = 4 4 Step 2 If f'(x) is negative, then f(x) is decreasing decreasing. If f'(x) is positive, then f(x) is increasing increasing Step 3 If 0
Consider the equation below. f(x) = 6 sin(x) + 6 cos(x), 0 sxs 2n Exercise (a) Find the interval on which f is increasing. Find the interval on which f is decreasing. Step 1 For f(x) = 6 sin(x) + 6 cos(x), we have f'(x) = 6 cos (x) – 6 sin (x 6 cos (x) – 6 sin(x) If this equals 0, then we have cos(x) = sin (x) sin(x) which becomes tan(x) = Hence, in the interval 0 s x s 27, f'(x) = 0 when x = or 57 X = 4 4 Step 2 If f'(x) is negative, then f(x) is decreasing decreasing. If f'(x) is positive, then f(x) is increasing increasing Step 3 If 0
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter5: Exponential And Logarithmic Functions
Section5.2: Applications Of Exponential Functions
Problem 48E
Related questions
Question
![Consider the equation below.
f(x) = 6 sin(x) + 6 cos(x),
0 < x < 2n
Exercise (a)
Find the interval on which f is increasing. Find the interval on which f is decreasing.
Step 1
For f(x) = 6 sin(x) + 6 cos(x), we have
f'(x) = 6 cos (x) – 6 sin(x)
6 cos (x) – 6 sin(x)
If this equals 0, then we have cos(x) = sin (r)
sin(x)
which becomes tan(x) = 1
1. Hence, in the interval 0 < x < 2n, f'(x) = 0 when x =
or
X =
4
4
Step 2
If f'(x) is negative, then f(x) is decreasing v
decreasing
If f'(x) is positive, then f(x) is increasing
increasing
Step 3
If 0 <x < , then f'(x) is positive
positive
and f(x) is increasing
increasing
Step 4
If 1 < x <
4
, then f'(x) is negative
negative,
and f(x) is decreasing
decreasing
Step 5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F435327ce-ad1d-47b0-9d05-42b05e19b7ae%2F7ca7db13-16f5-4ea4-b864-bd355912d7a0%2F4c0hsg_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the equation below.
f(x) = 6 sin(x) + 6 cos(x),
0 < x < 2n
Exercise (a)
Find the interval on which f is increasing. Find the interval on which f is decreasing.
Step 1
For f(x) = 6 sin(x) + 6 cos(x), we have
f'(x) = 6 cos (x) – 6 sin(x)
6 cos (x) – 6 sin(x)
If this equals 0, then we have cos(x) = sin (r)
sin(x)
which becomes tan(x) = 1
1. Hence, in the interval 0 < x < 2n, f'(x) = 0 when x =
or
X =
4
4
Step 2
If f'(x) is negative, then f(x) is decreasing v
decreasing
If f'(x) is positive, then f(x) is increasing
increasing
Step 3
If 0 <x < , then f'(x) is positive
positive
and f(x) is increasing
increasing
Step 4
If 1 < x <
4
, then f'(x) is negative
negative,
and f(x) is decreasing
decreasing
Step 5
![Step 6
Therefore, the interval on which f is increasing is
(-0,),(,∞0) (Enter your answer using interval notation.)
and the interval on which f is decreasing is
(Enter your answer using interval notation.).
Submit
Skip (you cannot come back)
Exercise (b)
Find the local minimum and maximum values of f.
Step 1
π
We know f(x) changes from increasing to decreasing at x =
Therefore,
is a
Select-- v
Submit Skip (you cannot come back)
Exercise (c)
Find the inflection points. Find the interval on which f is concave up. Find the interval on which f is concave down.
Click here to begin!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F435327ce-ad1d-47b0-9d05-42b05e19b7ae%2F7ca7db13-16f5-4ea4-b864-bd355912d7a0%2Fmibyw3a_processed.png&w=3840&q=75)
Transcribed Image Text:Step 6
Therefore, the interval on which f is increasing is
(-0,),(,∞0) (Enter your answer using interval notation.)
and the interval on which f is decreasing is
(Enter your answer using interval notation.).
Submit
Skip (you cannot come back)
Exercise (b)
Find the local minimum and maximum values of f.
Step 1
π
We know f(x) changes from increasing to decreasing at x =
Therefore,
is a
Select-- v
Submit Skip (you cannot come back)
Exercise (c)
Find the inflection points. Find the interval on which f is concave up. Find the interval on which f is concave down.
Click here to begin!
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