In my textbook, I have read that $f''(x)=0$ is a necessary but not sufficient condition for a point to be a point of inflection.
However, what happens at vertical tangents where the second derivative does not exist? Can these points be points of inflection even though $f''(x)= DNE$?
For instance, take $f(x)=x^{1/3}$. There is a vertical tangent when x equals 0... Is this a point of inflection as there is a change in concavity? If so, did the textbook omit/forget about this possibility?
$f''(x)=0$ is not a necessary condition.
Following two conditions must be met for inflection point to exist:
1) $f(x)$ must be continuous
2) Concavity must change, that means sign of $f''(x)$ must change
In this case $f(x)$ is continuous at $x = 0$
$f''(x) = -\dfrac{2}{9x^{5/3}}$, which changes sign at $x=0$
So $(0,0)$ is indeed an inflection point for $f(x)=x^{1/3}$