If the second derivative of a function is zero, why is the second derivative test inconclusive?

Question

2nd derivative test gives three possibilities: 1) greater than zero (strict local min) 2) less than zero (strict local max) 3) equal to zero - no information

It is this third case that I do not understand. If the second derivative at a stationary point is zero, doesn't that mean it is a stationary inflection point?

I understand that another criteria for an inflection point is that it must change signs. However, I cannot think of an example where the first derivative is zero and the second derivative is zero, but the graph does not change signs.

There is some information about undulation points available, but the descriptions are fairly vague: What is the difference between an undulation point and other critical values?

I understand why the second derivative is zero at an inflection point: Why is the second derivative of an inflection point zero?

I understand why the second derivative must change sign at an inflection point (the concavity/convexity changes).

Answer 1

For a casuistic example, consider $f(x) \equiv 0$. No inflection points here at all.

Answer 2

So if a double derivative is zero then there are two possibilities

1 there is inflection point 2 there is no point of inflection

Now if there is point of inflection at the given point then double derivative before than point and after that point must be of different signs and so if they are of same signs then there is point of inflection.