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Problem 22:
- One relative minimum at
- Two relative maxima at
and
Answer: A. one relative minimum and two relative maxima.
Problem 23:
- Points of inflection at
and
Answer: D.
,
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To determine the relative extrema of the function
, we can start by calculating its first derivative,
, and setting it to zero to find critical points. Factoring gives us
, leading to critical points at
. Evaluating the second derivative,
, allows us to confirm maximums at
and
, and a minimum at
. So, you have one relative minimum and two relative maxima (Answer B).
To find the points of inflection, we look for where the second derivative changes sign, starting from
. Setting it to zero leads to
, resulting in
and thus
and
. These points are where the concavity of the graph changes, providing the x-coordinates of the inflection points!