The function is
.
Rewrite the function is
.
Apply first derivative on each side with respect to
.
.
.
.
Apply second derivative on each side with respect to
.


.
Determine the values of
at which
or
does not exist.




and
and
.
,
and
.
The values of
are
,
and
.
Test for concavity in the intervals
,
,
and
.
| Intervals | \Test value | \ \
Sign of | \
Conclusion | \
![]() | \
\
| \
\
| \
\
Concave upward \ | \
![]() | \
![]() | \
![]() | \
Concave downward | \
![]() | \
\
| \
\
| \
\
Concave upward \ | \
![]() | \
\
| \
\
| \
Concave downward | \
The function is concave upward in the interval
and
.
The function is concave downward in the interval
and
.