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| PAGE: 229 | SET: Exercises | PROBLEM: 31 |
(a)
I/D test :
If 
on the interval, then 
is increasing on the interval.
If 
on the interval, then is decreasing on the interval.
Observe the graph.
over the intervals 
, 
and 
.
Therefore the function 
is increasing over the intervals 
, 
and 
.
over the intervals 
and 
.
Therefore the function 
is decreasing over the intervals 
and 
.
(b)
First derivative test :
(i) If 
changes from positive to negative at 
, then has a local maximum at .
(ii) If changes from negative to positive at , then has a local minimum at .
Observe the graph.
changes from positive to negative at 
and 
.
Therfore the function has a local maximum at 
and 
.
changes from negative to positive at 
and 
.
Therfore the function has a local minimum at 
and 
.
(c)
Concavity test :
If 
for all 
in the interval, then the graph of is concave upward on the interval.
If 
for all in the interval, then the graph of is concave downward on the interval.
Observe the graph.
is increases over the interval 
, then on the intervals.
Therefore the function has a concave upward over the intervals 
.
is decreases over the interval 
, then on the intervals.
Therefore the function has a concave downward over the intervals 
.
(d)
Inflection points :
Inflection points are the points at which the concavity changes from up to down or down to up.
Observe the graph.
At 
the function changes from decreasing to increasing, then 
changes from negative to positive.
Therefore the inflection point is 
.
(e)
The function is 
.
Graph :
Such that above all the conditions are satisfied :
(a)
The function is increasing on , and .
The function is decreasing on and .
(b)
The function has a local maximum at and .
The function has a local minimum at and .
(c)
The function has a concave upward on .
The function has a concave downward on .
(d) The inflection point is .
(e) Graph of the function is
.
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