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Q. 33

Expert-verifiedFound in: Page 261

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{l}\mathit{n}\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{)}$

Ans:

After inserting the root values we can find the increasing and decreasing intervals of the given function.

Intervals of the given function are

Increasing at $\left(0,\infty \right)$

Decreasing at$\left(-\infty ,0\right)$

$\mathit{l}\mathit{n}\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{)}$

$f\left(x\right)=\mathrm{ln}\left({x}^{2}+1\right)\phantom{\rule{0ex}{0ex}}{f}^{\text{'}}\left(x\right)=\frac{2x}{{x}^{2}+1}\phantom{\rule{0ex}{0ex}}let{f}^{\text{'}}\left(x\right)=0\phantom{\rule{0ex}{0ex}}\therefore \frac{2x}{{x}^{2}+1}=0\phantom{\rule{0ex}{0ex}}\Rightarrow 2x=\left({x}^{2}+1\right)0\phantom{\rule{0ex}{0ex}}\Rightarrow 2x=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}x=0\phantom{\rule{0ex}{0ex}}$

After inserting the root values we can find the increasing and decreasing intervals of the given function.

Intervals of the given function are

Increasing at $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{\infty}\mathbf{)}$

Decreasing at$\mathbf{(}\mathbf{-}\mathbf{\infty}\mathbf{,}\mathbf{0}\mathbf{)}$

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