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Q. 37

Expert-verified
Found in: Page 261

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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

Ans: Increasing interval $\mathbf{\left[}\mathbf{-}\frac{\mathbf{\pi }}{\mathbf{4}}\mathbf{+}\mathbf{\pi k}\mathbf{,}\frac{\mathbf{\pi }}{\mathbf{4}}\mathbf{+}\mathbf{\pi k}\mathbf{\right]}$

and decreasing elsewhere.

See the step by step solution

## Step 1. Given information:

$\mathbit{f}\mathbf{\left(}\mathbit{x}\mathbf{\right)}\mathbf{=}\mathbit{s}\mathbit{i}\mathbit{n}\mathbit{x}\mathbf{.}\mathbit{c}\mathbit{o}\mathbit{s}\mathbit{x}$

## Step 2. Finding the derivative of the function:

$f\left(x\right)=\mathrm{sin}x.\mathrm{cos}x\phantom{\rule{0ex}{0ex}}{f}^{\text{'}}\left(x\right)=\mathrm{sin}x\left(-\mathrm{sin}x\right)+\mathrm{cos}x\mathrm{cos}x\phantom{\rule{0ex}{0ex}}{f}^{\text{'}}\left(x\right)={\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\phantom{\rule{0ex}{0ex}}{f}^{\text{'}}\left(x\right)=\mathrm{cos}2x\phantom{\rule{0ex}{0ex}}let{f}^{\text{'}}\left(x\right)=0\phantom{\rule{0ex}{0ex}}\therefore \mathrm{cos}2x=0\phantom{\rule{0ex}{0ex}}⇒2x=\left(2k+1\right)\frac{\pi }{2}\phantom{\rule{0ex}{0ex}}⇒x=\left(2k+1\right)\frac{\pi }{4}\left[wherekisanyinteger\right]\phantom{\rule{0ex}{0ex}}takingpointx=0$

## 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 $\mathbf{\left[}\mathbf{-}\frac{\mathbf{\pi }}{\mathbf{4}}\mathbf{+}\mathbf{\pi k}\mathbf{,}\frac{\mathbf{\pi }}{\mathbf{4}}\mathbf{+}\mathbf{\pi k}\mathbf{\right]}$

and decreasing elsewhere.