The article deals with the problems leading to finding the extremum of a linear function. The purpose of the article is to show a general method for solving extreme problems related to a linear function and to reveal the significance of this method in the earlier acquisition of knowledge about extreme problems from students. The methodological basis of the research is a systematic approach to teaching secondary school students to solve problems at extremes. The following results were obtained in the study: the concept of "extremum problems" was defined, the structure of extremum problems associated with a linear function was considered, a general method for solving problems that reduce to finding the extremum of a linear function was developed, the stages of work on such problems were shown. It is noted that this method does not require calculating the derivative of the function and it is based primarily on the monotonicity of a linear function on a certain segment. Therefore, a modern student should not only be able to solve extreme problems using a derivative, but also possess elementary methods.