$f^{\prime\prime}(x) = 0$ and $f^{\prime\prime}(x) = \text{DNE}$ will identify CANDIDATES for points of inflection.
To determine if a candidate is the location of the Point of Inflection, you could either:
1. Evaluate $f^{\prime\prime}(x)$ before and after $x =$ candidate. If there is a sign change, then the candidate is the $x$-value of a POI.
2. Evaluate $f^{\prime\prime\prime}$(candidate). If the 3rd-derivative does not equal zero, then the candidate is the $x$-value of a POI.

Point of Inflection at $\left(0,0\right)$.
Be sure to show all steps and justification/tests.