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Q. 68

Found in: Page 276


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', and f'', and examine any relevant limits so that you can describe all key points and behaviors of f.


The sketch of the graph is

The sign chart is

See the step by step solution

Step by Step Solution

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.



Let's examine the limits

limxf(x)=(1-x)4-2 limxf(x)=Andlimx-f(x)=(1-x)4-2limx-f(x)=

Step 3. Testing the signs. 

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


role="math" localid="1648472481521" f'(x)=-41-x30=-41-x30-4 and 1-x3=0 x=1

Testing the signs on both sides,


Thus, f' is positive on the interval 1, and negative on the interval -,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.


f''(x)=121-x20=121-x2012 and 1-x2=0 x=1

Testing the signs on both sides,


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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