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| PAGE: 221 | SET: Exercises | PROBLEM: 9 |
(a)
The function is 
.
Differentiate 
on each side with respect to 
.
Find the critical points.
Since it is a polynomial it is continuous at all the point.
Thus, the critical points exist when 
.
Equate 
to zero.
and 
and 
.
The critical points are 
and 
.
The test intervals are 
.
| Interval | Test Value | Sign of  |
Conclusion |
 |
 |
 |
Increasing |
 |
 |
 |
Decreasing |
 |
 |
 |
Increasing |
Therefore the function is increasing on the intervals 
and 
.
The function is decreasing on the interval 
.
(b)
Find the local maximum and local minimum.
The function 
has a local maximum at , because 
changes its sign from positive to negative.
Substitute in 
.
Local maximum is 
.
The function has a local maximum at , because changes its sign from positive to negative.
Local minimum is 
.
(c)
.
Differentiate on each side with respect to 
.
Find the inflection points.
Equate 
to zero.
The inflection point is at 
.
Substitute 
in .
The test intervals are 
and 
.
|
Interval |
Test Value | Sign of |
Concavity |
|
 |
 |
Down |
|
|
 |
Up |
The graph is concave up on the interval .
The graph is concave down on the interval .
The inflection point is 
.
(a)
Increasing on the intervals and .
Decreasing on the interval .
(b)
Local maximum is .
Local minimum is
(c)
Concave up on the interval .
Concave down on the interval .
Inflection point is 
.
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