CPM Homework Help : CALC Problem 6-171

Thoroughly investigate the graph of $f(x) = x^{4/3} + 2x^{1/3}$ . Identify all important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify any point(s) of inflection and intercepts, and provide graphs of $f^\prime(x)$ and $f^{\prime\prime}(x)$ . Be sure to justify all statements both graphically and analytically.

Hint 1:

Remember to investigate all extrema CANDIDATES. That means, find all values of $x$ where $f^\prime(x) = 0$ AND f '(x) DNE.
After all, a max or min could exist at a cusp... But remember, sometimes these candidates turn out to be inflection points instead.

Hint 2:

When looking for points of inflection, remember to consider where concavity changes. This could happen where $f^{\prime\prime}(x) = 0$
or where $f^{\prime\prime}(x) =$ DNE.