My mnemonic device for the Second Derivative Test (note what happens in the zero case):

It could be concave up, or concave down, or a point of inflection. Pretty confused individual! All this to say, I don't think you can know if you have a point of inflection there or not. The function has its second derivative zero at the origin, but the origin is not a point of inflection: it's concave up there. Similarly, is concave down at the origin - no point of inflection. Finally, does have a point of inflection at the origin.