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| PAGE: 404 | SET: Exercises | PROBLEM: 10 |
(a)
The function 
.
Find the 
-intercept:
Consider 
.
Find the -intercept, by substituting 
in .
.
The -intercept is 
.
(b)
Find for what numbers the graph of 
is decreasing in the interval is 
.
The function .
Apply derivative on each side with respect to 
.
Determination of critical points :
The critical points exist when 
.
Equate 
to zero:
Solve in the interval .
General solution of 
is 
, where 
is an integer.
General solution is
If 
, .
If 
, 
.
If 
,
The solutions are 
,
and 
in the interval .
The critical points are , and and the test intervals are 
.
| Interval | Test Value | Sign of |
Conclusion |
 |
 |
|
Increasing |
 |
 |
|
Decreasing |
The function is Decreasing over the interval .
(c)
The function and the interval is .
The critical points are and and 
.
Find the values of 
at these critical points.
.
.
.
Compare the values of to find the absolute minimum.
Absolute minimum value is 
.
(d)
The function and the interval is 
.
General solution of 
is 
, where is an integer.
General solution: .
If , 
.
If , 
.
If 
, 
.
The solutions are and 
in the interval .
(e)
The function and the interval is 
.
General solution of 
is , where is an integer.
General solution: .
If , 
.
If , 
.
If , 
.
The solutions are , and 
in the interval .
General solution of is , where is an integer.
General solution: .
If , 
.
If , 
.
The solutions are and in the interval .
(f)
General solution of is , where is an integer.
General solution: .
If , 
.
If , 
.
If , 
.
The solutions are 
in the interval .
(g)
Find the -intercept :
The function 
.
Find the -intercept, by substituting 
in .
General solution of is 
, where is an integer.
The -intercepts are , where is an integer.
(a). The -intercept is .
(b).The function is Decreasing over the interval .
(c). Absolute minimum value is 
.
(d).The solutions are and in the interval .
(e). For 
,the solutions are , and in the interval
For 
, the solutions are and in the interval .
(f). The solutions are in the interval .
(g).The -intercepts are , where is an integer.
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