### Home > CALC > Chapter 5 > Lesson 5.1.3 > Problem5-30

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.

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.

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