For a max or min, it's necessary that the first derivative is zero, and then that the first non-vanishing derivative is an even numbered one. If that derivative is negative then the point will be a max, if it's positive then the point is a min. If the first non-vanishing derivative is an odd numbered one, then you have a point of inflection, (so it is not sufficient simply to evaluate the second, fourth, ... derivatives, you must evaluate the odd numbered ones as well). (Unless that is you know for certain that the point is definitely a max or min). Usually you will not know in advance just how far you have to keep differentiating.

The (somewhat disgusting) function you give as an example, were you given it or did you make it up ?

Any differentiation for this function is overkill. It's even, meaning that its graph will be symmetric about the vertical axis. All you need do is evaluate it at the origin and some near-by point and compare values.