Second Derivative Test Conditions

Question

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?

Answer 1

$f''(x)=0$ 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

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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 $f(x)=x^{1/3}$