In the school course study of algebra and the beginning of analysis, the main focus is on teaching students to study functions using differential calculus, that is, to study function by a new method - the method of mathematical analysis. The development of functional representations helps high school students get visual ideas about the continuity of a function in the area of ​ ​ its definition, learn to build their graphs. This is due to the fact that the concept of function is widely used in different branches of mathematics, in solving various applied problems. The paper examines various techniques for investigating the largest and smallest values of some algebraic and transcendental functions without using a derivative. The ability to apply various methods of studying functions can contribute to the successful assimilation of elements of mathematical analysis, the development of research skills, the interest of students in conducting scientific research and, in general, increase interest in mathematics.