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| PAGE: 403 | SET: Exercises | PROBLEM: 9 |
(a)
The function 
.
Find the 
-intercept:
Find the 
-intercept, by substituting 
in 
.
.
The -intercept is 
.
(b)
The function 
and the interval is 
.
Differentiate the function 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 
, 
.
The solutions are 
in the interval .
The critical points are 
and the test intervals are 
.
| Interval | Test Value | Sign of  |
Conclusion |
 |
 |
|
Decreasing |
 |
|
|
Increasing |
 |
 |
|
Decreasing |
The function is increasing over the interval 
.
(c)
The function and the interval is .
The critical points are .
Find the values of 
at these critical points.
.
.
Find the values of at the end points of the interval.
.
.
Compare the four values of to find the absolute maximum.
Absolute maximum 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 
, 
.
If , 
.
The solution is 
and 
in the interval 
.
General solution of is , where is an integer.
General solution: 
.
If , 
.
If , 
.
If , 
.
If 
, 
.
The solution is 
and in the interval .
(f)
General solution of is , where is an integer.
General solution: 
.
If , 
.
If , 
.
If 
, 
.
If , 
.
If 
, 
.
If 
, 
.
The solutions are 
and 
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 increasing over the interval .
(c). Absolute maximum 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 and in the interval .
(g).The - intercepts are , where is an integer.
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