(a)
\The function is .
Find the critical numbers by applying derivative.
\Apply derivative on each side with respect to .
Equate the derivative to .
The function has a discontinuity at .
Therefore critical number is .
(b)
\The function has a discontinuity at .
So we need consider the discontinuity while setting the test intervals.
\Consider the test intervals to find the interval of increasing and decreasing.
\Test intervals are ,
and
.
Test interval | Test value | Sign of ![]() | Conclusion |
![]() | ![]() | \
| Decreasing |
![]() | ![]() | \
| Increasing |
![]() | ![]() | \
| Decreasing |
The function is increasing on the interval
.
The function is decreasing on the intervals
and
.
(c)
\Use first derivative test to identify all relative extrema.
\ changes from negative to positive at
. [From (b) ]
Therefore according to first derivative test , the function has minimum at .
The function has a relative minimum at
.
Find .
So the function has relative minimum at
.
(d)
\Graph the function is .
Now observe the graph :
\Critical number is .
The function has a discontinuity at .
The function is increasing on the interval
.
The function is decreasing on the intervals
and
.
The function has relative minimum at
.
(a)
\Critical number is .
The function has a discontinuity at .
(b)
\The function is increasing on the interval
.
The function is decreasing on the intervals
and
.
(c)
\The function has relative minimum at
.
(d)
\Graph of the function is .