Application of Methods of Mathematical Analysis to Prove Olympiad Inequalities

Abstract

The high requirements regarding the content of the knowledge, abilities and skills, which determines the capacity of the specialist to compete on the modern labour market are set to nowadays graduates. The tasks which require not only the knowledge of school curriculum, but also the creative application of this knowledge, in particular for inequalities solving are reviewed during Math’s course. This issue is quite relevant, because the tasks of this type are found in the tasks of school, district math Olympiads. Inequalities take a significant part of the school mathematics’ course. Applied tasks are written into Math’s language with the help of inequalities. In addition, inequalities are a tool that allows to repeat, fix, deepen the theoretical knowledge in each subject and to develop creative mathematical capacity. This topic contains many ways, methods of solving them and methods of proving them. Proof of inequalities must be given special attention because it plays an important role in shaping the logical thinking and mathematical culture. Tasks for proving inequalities make it possible to consolidate a wide range of theoretical issues studied in the school course of mathematics (theory of inequalities, properties of functions, questions of equivalent equations), they encourage the formation of critical thinking, the ability to ground actions logically. In addition, knowledge of classical inequalities and methods of proving them gives the opportunity to apply inequalities more widely in solving other problems, including applications. Since the tasks of proving inequalities are very diverse, there is no single general way to prove any inequality. Proving inequalities has a significant impact on the formation and development of creative thinking and creative personality of the student due to the availability of different ways to prove inequality. Different methods of inequalities solving are considered in this article. The peculiarities of pupils’ preparation by the method of proving contest and Olympiad inequalities, such as f12.jpg with the fixed sum of variables, are considered in this article. Let’s review the peculiarities of differential count set usage on the level of senior pupil. The ways of proving the inequalities with tangent or n-1 statement of equal meanings are analyzed, their advantages and disadvantages are reviewed. With the help of these notions it is possible to algorithmizate the process of proving several kinds of inequalities. Several ways of proving are introduced for some kinds of tasks, such methods of inequalities solving demand from pupils the basic knowledge in differential counting. Solving such problems contributes to intellectual development, the development of logical thinking and is a good material for the development of skills.