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Q. 37

Expert-verified

Calculus

Found in: Page 261

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Use a sign chart for $f^{'}$ to determine the intervals on which each function $f$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.
role="math" localid="1648370582124" $f (x) = s i n x . c o s x$

Ans: Increasing interval $[- \frac{π}{4} + πk, \frac{π}{4} + πk]$

and decreasing elsewhere.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information:

$f (x) = s i n x . c o s x$

Step 2. Finding the derivative of the function:

$f (x) = \sin x . \cos x f^{'} (x) = \sin x (- \sin x) + \cos x \cos x f^{'} (x) = \cos^{2} x - \sin^{2} x f^{'} (x) = \cos 2 x l e t f^{'} (x) = 0 ∴ \cos 2 x = 0 \Rightarrow 2 x = (2 k + 1) \frac{π}{2} \Rightarrow x = (2 k + 1) \frac{π}{4} [w h e r e k i s a n y i n t e g e r] t a k i n g p o i n t x = 0$

Step 3. Finding increasing and decreasing intervals:

Intervals of the given function :
f'(x) has x=0f'(0)=cos(2.0) =cos0 =1 >0

f(x)is increasing on the interval $[- \frac{π}{4} + πk, \frac{π}{4} + πk]$

and decreasing elsewhere.

Step 4. Verifying algebraic answers with graphs :

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