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

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).

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