Can someone tell me what this actually is. So, in the case when the Hessian is positive (or negative) semidefinite, the second derivative test is inconclusive. However, I think I've read that even in the case where the Hessian is positive semidefinite at a stationary point x, we can still conclude that the function at x is not a local maximum. Is that correct? Is that equivalent to the function at x being either a local minimum or a saddle point, since there are only 3 possibilities for x: local max, local min, or saddle (or are there more possibilties)? If someone has some online pdf notes with definitions and proof of the second derivative test in generality, that would be good too. I also can't seem to find an agreed on definition of a saddle point, which adds to the confusion.