Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f, f', a n d f'',$ and examine any relevant limits so that you can describe all key points and behaviors of f.
$f (x) = {(1 - x)}^{4} - 2$

The sketch of the graph is

The sign chart is

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information.

The given function is $f (x) = {(1 - x)}^{4} - 2 .$

Step 2. Finding the roots and examining the relevant limit.

To find the roots we will put the given function equal to zero.

So,

$f (x) = {(1 - x)}^{4} - 2 0 = {(1 - x)}^{4} - 2$

Let's examine the limits

$\lim_{x \to \infty} f (x) = {(1 - x)}^{4} - 2 \lim_{x \to \infty} f (x) = \infty A n d \lim_{x \to - \infty} f (x) = {(1 - x)}^{4} - 2 \lim_{x \to - \infty} f (x) = \infty$

Step 3. Testing the signs.

To sketch the sign chart, let's differentiate the equation to find $f' .$

So,

role="math" localid="1648472481521" $f' (x) = - 4 {(1 - x)}^{3} 0 = - 4 {(1 - x)}^{3} 0 \neq - 4 a n d {(1 - x)}^{3} = 0 x = 1$

Testing the signs on both sides,

$f' (0) = - 4 {(1 - 0)}^{3} f' (0) = - 4 A n d f' (2) = - 4 {(1 - 2)}^{3} f' (2) = 4$

Thus, $f'$ is positive on the interval $(1, \infty)$ and negative on the interval $(- \infty, 1) .$ Hence the graph of f will be increasing on the positive intervals and decreasing on the negative intervals.

Step 4. Testing the signs.

Now, let's test the sign for $f'' .$

Let's differentiate again.

So,

$f'' (x) = 12 {(1 - x)}^{2} 0 = 12 {(1 - x)}^{2} 0 \neq 12 a n d {(1 - x)}^{2} = 0 x = 1$

Testing the signs on both sides,

$f'' (0) = 12 {(1 - 0)}^{2} f'' (0) = 12 A n d f'' (2) = 12 {(1 - 2)}^{2} f'' (2) = 12$

Thus, $f''$ is positive. Hence f will be concave up everyplace.

Step 5. Sketch the sign chart.

The sign chart is

Step 6. Sketch the graph of function f.

The graph of the function is

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Calculus

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Short Answer

Step by Step Solution

Step 1. Given Information.

Step 2. Finding the roots and examining the relevant limit.

Step 3. Testing the signs.

Step 4. Testing the signs.

Step 5. Sketch the sign chart.

Step 6. Sketch the graph of function f.

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Calculus

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Step by Step Solution

Step 1. Given Information.

Step 2. Finding the roots and examining the relevant limit.

Step 3. Testing the signs.

Step 4. Testing the signs.

Step 5. Sketch the sign chart.

Step 6. Sketch the graph of function f.

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