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| PAGE: 243 | SET: Exercises | PROBLEM: 15 |
The function is 
.
(A)
Domain :
The function is 
.
All possible values of 
is the domain of the function.
Denominator of the function should not be zero.
Here denominator of the function 
.
Therefore, 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 -intercept is .
(C)
Symmetry :
Substitute 
in the function.
Hence 
.
Therefore the function 
is an odd function.
(D)
Asymptotes :
Horizontal asymptote :
Therefore the horizontal asymptote is 
.
Vertical asymptote :
Vertical asymptote appears when the function is not defined.
The function is defined for all real numbers.
Therefore, there are no vertical asymptotes.
(E)
Intervals of increase or decrease :
The function is .
Differentiate on each side with respect to .
.
Find critical points by equating 
.
and 
.
Test intervals are 
,
and 
.
|
Interval |
Test Value | Sign of  |
Conclusion |
 |
 |
|
Decreasing |
 |
 |
|
Increasing |
 |
 |
|
Decreasing |
The graph is increasing in the interval .
The graph is decreasing in the interval and .
(F)
Local Maximum and Minimum values :
The function 
has a local minimum at , because 
changes its sign from negative to positive.
Substitute 
in 
.
.
Local minmum is 
.
The function has a local maximum at , because changes its sign from positive to negative.
Substitute in .
.
Local maximum is 
.
(G)
Concavity and point of inflection :
.
Differentiate 
on each side with respect to .
.
Find inflection point by equating 
to zero.
and 
, 
and 
.
Inflection points are , and .
Test intervals into 
, 
, 
and 
.
|
Interval |
Test Value | Sign of |
Concavity |
 |
 |
|
Down |
 |
 |
|
Up |
 |
|
|
Down |
 |
 |
|
Up |
The graph is concave up in the interval and .
The graph is concave down in the interval and .
(H)
Graph :
Graph of the function :
(A) The domain of the function 
is .
(B) -intercept is and -intercepts are .
(C) The function is an odd function.
(D) The horizontal asymptote is and there is no vertical asymptotes.
(E) The graph is increasing in the interval 
.
The graph is decreasing in the interval 
.
(F)
Local minmum is .
Local maximum is .
(G)
The graph is concave up in the interval and .
The graph is concave down in the interval and .
(H) Graph of the function is
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