General Derivative Test Flow Chart

I was teaching my students today how one could use higher derivates to determine the shape of a curve at a particular point. The problem that really can be confusing is the case when $$ y ‘ = y ‘ ‘ = 0. $$ I felt that a flow chart with pictures would be the best way to represent the process for deciding what the shape is. That is, telling whether or not you have a min, max, inflection, saddle, etc etc etc. So, good old MS paint to the rescue 🙂