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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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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
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!
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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