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Q .28

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.$\mathbit{f}\mathbf{\left(}\mathbit{x}\mathbf{\right)}\mathbf{=}{\mathbit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{4}{\mathbit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathbit{x}\mathbf{-}\mathbf{5}$

Ans :

Intervals of the given function is $\mathbf{\left(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{\right]}\mathbf{\cup }\mathbf{\left[}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{,}\mathbf{\infty }\mathbf{\right)}$

Increasing at $\mathbf{\left(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{\right]}\mathbf{\cup }\mathbf{\left[}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{,}\mathbf{\infty }\mathbf{\right)}$$\mathbf{\left(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{\right]}\mathbf{\cup }\mathbf{\left[}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{,}\mathbf{\infty }\mathbf{\right)}$

Decreasing at $\mathbf{\left(}\mathbf{-}\mathbf{2}\mathbf{,}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{\right)}$

See the step by step solution

## Step 1. Given information:

$\mathbit{f}\mathbf{\left(}\mathbit{x}\mathbf{\right)}\mathbf{=}{\mathbit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{4}{\mathbit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathbit{x}\mathbf{-}\mathbf{5}$

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

$f\left(x\right)={x}^{3}+4{x}^{2}+4x-5\phantom{\rule{0ex}{0ex}}{f}^{\text{'}}\left(x\right)=3{x}^{2}+8x+4\phantom{\rule{0ex}{0ex}}let,{f}^{\text{'}}\left(x\right)=0\phantom{\rule{0ex}{0ex}}\therefore 3{x}^{2}+8x+4=0\phantom{\rule{0ex}{0ex}}3{x}^{2}+6x+2x+4=0\phantom{\rule{0ex}{0ex}}3x\left(x+2\right)+2\left(x+2\right)=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{)\overline{)3x+2=0\phantom{\rule{0ex}{0ex}}x=\frac{-2}{3}}}\phantom{\rule{0ex}{0ex}}\overline{)x+2=0\phantom{\rule{0ex}{0ex}}x=-2}\phantom{\rule{0ex}{0ex}}x=\frac{-2}{3},-2$

## Step 3. Inserting the root points on the number line(Sign chart):

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

Intervals of the given function is $\mathbf{\left(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{\right]}\mathbf{\cup }\mathbf{\left[}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{,}\mathbf{\infty }\mathbf{\right)}$

Increasing at $\mathbf{\left(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{\right]}\mathbf{\cup }\mathbf{\left[}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{,}\mathbf{\infty }\mathbf{\right)}$

Decreasing at$\mathbf{\left(}\mathbf{-}\mathbf{2}\mathbf{,}\frac{\mathbf{-}\mathbf{2}}{\mathbf{3}}\mathbf{\right)}$