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(a)
Thu 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.
and .
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 .
(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 minimum at , because changes its sign from negative to positive.
Local minimum is .
(c)
.
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 on the interval .
The graph is concave down on the interval .
The inflection point is .
(d)
Graph :
Graph the function :
(a)
Increasing on the intervals and .
Decreasing on the interval .
(b)
Local maximum is .
Local minimum is .
(c)
Concave up in the interval .
Concave down in the interval .
Inflection point is .
(d)
Graph of the function is
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