CPM Homework Help : APCALC Problem 6-170

Thoroughly investigate the graph of $f(x) = x^{4/3} + 2x^{1/3}$ . Identify all of the 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 $y = f^\prime(x)$ and $y = f^{\prime\prime}(x)$ . Be sure to justify all statements both graphically and analytically.

Hint:

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

Hint:

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