I would like to know how to correctly define and classify turning points using elementary calculus. The points I wish to clarify are maxima, minima, inflection points and saddle points. So I am aware of the basic info available everywhere, such as that a point is a maximum if and only if the first derivative is 0 and second derivative is negative there, and a minimum if and only if the first derivative is 0 and second derivative is positive there. Sites are much less clear on how to determine (or even define) inflection points and saddle points for certain. Furthermore, I believe the first two pieces of information are insufficient - if the second derivative is 0, it is also possible the point is a minimum or maximum (e.g. y=x4). Can I get an explanation of what analysis to do in order to classify the turning points correctly?