CPM Homework Help : CALC Problem 5-30

Mr. Lyon was trying to find the points of inflection for $f(x) = 3x^{\frac{2}{3}}$ . Since no value of $x$ existed such that $f^{\prime\prime}(x) = 0$ , he assumed there was no change in concavity. Andrew said that a point of inflection exists where the second derivative changes sign. Who is correct and why? Use the second derivative and the graph of $f(x)$ to verify your answer.

Hint 1:

CANDIDATES for points of inflection can be found where $f^{\prime\prime}(x) = 0$ and where $f^{\prime\prime}(x) =$ DNE. You must check both.

Graph:

Hint 2:

Notice that there is not a concavity change at the cusp.