About one remark in the textbook of mathematical analysis

The article discusses the tasks for applying the theorem on a sufficient conditions of the extremum of a function of both one variable and two variables. The question is to examine the sign of the second derivative at the point of a possible extremum. A comment is given on a remark in a classic textbook of mathematical analysis, which talks about the mistakes of famous mathematicians on the topic of a sufficient sign of the extremum of a function of two variables.