If the book says that the stagnation point is not necessarily the extreme point, but the extreme point must be the stagnation point. This statement is not rigorous. Strictly speaking, the stagnation point is not necessarily the extreme point, and the derivative extreme point must be the stagnation point. This means the existence of non-derivative extreme points. Therefore, we should find the extreme point at the stagnation point and make up lessons in the shop.

In fact, textbooks may not say that "stagnation is not necessarily the extreme point, but the extreme point must be stagnation", which may be some people's own understanding. As far as I know, the narration in the textbook is quite rigorous. For example, under the theorem 155 in the sixth edition of advanced mathematics of Tongji University, it is clearly pointed out that "the extreme point of the differentiable function f(x) must be her stagnation point". The second edition of Calculus edited by Zhu clearly points out on page 102 that "the extreme point of f(x) can only be located at the point where f'(x) does not exist and at the point where f'(x)= 0". Page 184 of the first volume of Advanced Mathematics edited by Liu Zhongkan said: "In the case of derivative, the extreme point must be the stagnation point, but the stagnation point is not necessarily the extreme point." Page 185 specifically pointed out: "The extreme value of a function may also be obtained at the non-derivative point." If your book really says so, you can suspect that it is a pirated book, or a fake that someone spelled correctly to evaluate the title.