When you have f'(x) = 0 and f''(x) = 0 for the same x, is it both a point of inflection and a max/min or just a max/min? I think it's only a max or min right? And x is not a point of inflection for f.
Do you mean there is a function where f"(c)=0 and f'(c)=0 but (1) is not a point of inflection, but is a max/min? or a point where both equal 0 and (2) there is a point of inflection and a max/min? or (3) there is point of inflection but no max/min.
(1) i can not seem to find an example for this but it does seem entirely possible
(2) in order for this situation to occur f(x) must come to a point, c, where the slope changes from increasing to decreasing, or vice-versa, and from concave up to concave down, or vice versa. In order for all this to happen at the point c, f must come to a sharp point. If f comes to a sharp point it is not differentiable. Thus the second is not a possibility.
(3) example of x^3 at x=0