My teacher told me (and Wikipedia backs this up) that the second derivative of a stationary point doesn't tell us anything about whether the point is a maximum, minimum or an inflection point. But I don't understand how it can be anything other than an inflection point.

Please can you:

a) explain why it can be a maximum/minimum with a second derivative of $0$

b) give an example where this happens (where there is a maximum/minimum and it has a second derivative of $0$)

Thank you.