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 | \
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\
| \
\
| \
\
Concave upward \ | \
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Concave downward | \
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\
| \
\
| \
\
Concave upward \ | \
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\
| \
\
| \
Concave downward | \
The function is concave upward in the interval and
.
The function is concave downward in the interval and
.