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Thoroughly investigate the graph of for . Identify all of the important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify point(s) of inflection and intercepts, and provide graphs of and . Be sure to justify all statements graphically and analytically.

To find INCREASING vs. DECREASING: Solve and .
To find CONCAVE UP vs. CONCAVE DOWN: Solve and .

LOCAL MAXIMA happen where increasing changes to decreasing.
LOCAL MINIMA happen where decreasing changes to increasing.
POINTS OF INFLECTION happen where concavity changes.

To find the GLOBAL MAXIMUM: compare with . The highest value wins.
To find GLOBAL MINIMUM: compare with . The lowest value wins.