since the objective of this one was to use the criterion of the first derivative.

My comment was trigger by exactly that thought. As I said, it can be misleading if the second is zero as well. In that case you might think you found an extreme while you are actually hitting on a saddle. While at it I added the discussion point about the difference between max and min.

Sorry if that was too exhausting, once I get triggered by math I cannot resist :-P

!BBH