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| PAGE: 242 | SET: Exercises | PROBLEM: 1 |
The function is 
.
(A)
Domain :
The function is 
.
All possible values of 
is the domain of the function.
The function 
is a polynomial function hence it is continuous for all the points.
Therefore the domain of the function is the set of all real numbers.
The domain of the function 
is 
.
(B)
Intercepts :
To find the 
-intercepts, substitute 
in the function.
Therefore the -intercept is 
.
To find the 
-intercepts, substitute in the function.
Therefore the -intercepts are and 
.
(C)
Symmetry :
Substitute 
in the function.
Here 
Therefore the function 
is neither odd nor even.
(D)
Asymptotes :
There is no vertical asymptotes, since the function is continuous at all real numbers.
Horizontal asymptote :
There is no horizontal asymptote.
(E)
Intervals of increase or decrease :
The function is .
Differentiate on each side with respect to .
Find the critical points.
Since 
is a polynomial it is continuous at all the point.
Thus, the critical points exist when 
.
Equate 
to zero.
The critical points are 
and 
.
The test intervals are 
, 
and 
.
| Interval | Test Value | Sign of |
Conclusion |
| |
 |
|
Increasing |
| |
 |
|
Decreasing |
| |
 |
|
Increasing |
The function is increasing on the intervals and 
.
The function is decreasing on the interval 
.
(F)
Local Maximum and Minimum values :
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 minimum at 
, because changes its sign from negative to positive.
Substitute in .
Local minimum is 
.
(G)
Concavity and point of inflection :
.
Differentiate 
on each side with respect to .
Find the inflection points.
Equate 
to zero.
The inflection point is 
.
Substitute 
in .
The test intervals are 
and 
.
|
Interval |
Test Value | Sign of  |
Concavity |
|
|
|
Down |
|
 |
|
Up |
The graph is concave up in the interval 
.
The graph is concave down in the interval 
.
The inflection point is 
.
(H)
Graph :
Graph of the function 
:
(A) Domain of the function is .
(B) -intercept is . and -intercepts are and 
.
(C) No symmetry.
(D) No asymptotes.
(E) Increasing on and .
Decreasing on .
(F) Local maximum is 
.
Local minimum is 
.
(G) Concave up on .
Concave down on .
Inflection point is .
(H) Graph of the function 
is
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