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Q. 69

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.

f(x) =x3(x+2)

The sign chart is

The sketch of the graph is

See the step by step solution

Step by Step Solution

Step 1. Given Information. 

The given function is f(x)=x3(x+2).

Step 2. Finding the roots.  

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


f(x)=x3(x+2)0=x3(x+2)x=0 and x+2=0 x=-2

Therefore, the given function have roots at x=0, -2.

Step 3. Testing the signs of f.  

To sketch the sign chart, let's test the signs on both sides.

For f

f(-3)=-33-3+2f(-3)=27Now, f(-1)=-13-1+2 f(-1)=-1Now, f(1)= 131+2 f(1)=3

Step 4. Testing the signs.  

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

Let's differentiate the equation to find f'.


f'(x)=4x3+6x20=4x3+6x20=2x22x+3x=0 and 0=2x+3 x=-32

Testing the signs on both sides,

f'(-2)=4-23+6-22f'(-2)=-8Now, f'(-1)=4-13+6-12f'(-1)=2Now, f'(1)=413+612f'(1)=10

Thus, f' is negative on the interval -,-32 and positive on the interval -32,. Hence the graph of f will be increasing on the positive intervals and decreasing on the negative intervals.

Let's differentiate again.


f''(x)=12x2+12x0=12xx+10=x and x+1=0 x=-1

Testing the sign on both sides,

role="math" localid="1648474789089" f''(-2)=12-22+12-2f''(-2)=24And f''(-0.5)=12-0.52+12-0.5f''(-0.5)=-3And f''(1)=1212+12(1)f''(1)=24

Thus, f'' is positive on the interval role="math" localid="1648474872810" -,-1 and 0, and negative on the interval -1,0. Hence, the graph of f will be concave up on the positive interval and concave down on the negative interval. Inflection point at x=-1,0.

Step 5. Sketch the sign chart. 

The sign chart is

Step 6. Examine the relevant limit.

Let's examine the limits of f(x)=x3x+2 as x±.


Step 7. Sketch the graph of function f.  

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