First of all, the extreme point is the local property of the function. Specifically, if the value of the function at this point is compared with other values in the small neighborhood of this point, the maximum or minimum value is obtained, and the corresponding maximum and minimum values are obtained. This concept has nothing to do with the derivability of the function itself. But for ordinary differentiable functions, the point where the first derivative is zero is often an extreme point, but it is not absolute. For example, if f (x) = x 3 and x=0, it is not an extreme point. Generally speaking, the point where f'=0 is called stagnation point, and there are only two extreme points, either stagnation point or non-derivative point. On the contrary, it is wrong. Non-derivative points or stagnation points are not necessarily extreme points.

Secondly, the inflection point is the point where the convexity and concavity of the function image (some textbooks call it convexity and concavity) changes, so it is called the inflection point, which has no essential connection with the extreme point and reflects two different mathematical properties. Similar to extreme points, inflection points consist of two kinds of points: one is the point where the second derivative is zero, and the other is the point where the second derivative does not exist.

The above part is what I have answered. If you can understand, all your questions can be explained. As for your supplementary question, I don't think it makes any sense. It's still a false proposition.