Creating new problems on proving inequalities, finding maximum and minimum values based on the critical properties and tangent inequalities of convex and concave functions

Tóm tắt

In this paper, we present some ideas and methods to create new problems of proving inequalities, problems of finding maximum and minimum values. Based on the maximum and minimum properties and tangent inequalities of convex and concave functions, we propose some ideas and methods to create new problems. We make all ideas and methods to be real via many specific functions. Especially, we combine the ideas and methods with equivalent transforms, Cauchy-Schwarz inequality, and inequality of arithmetic and geometric means to create new hard problems. New proposed examples, they have showed that our ideas and methods are important and efficient to lecturers at high schools and universities in giving questions in examinations, especially in examinations of selecting good students at levels, in Olympic examinations for high school and university students.