Using algebraic inequalities to solve extremum problems

The use of nonroutine tools in mathematical problem-solving is very beneficial: it encourages creativity, enables approaches that are easier and faster, and allows simplification instead of a tedious work. In this paper we present one such situation: we discuss the use of algebraic tools in the solution of extremum problems. These questions, which at first seem to have one classical solution, in fact can be solved using the approach of algebraic tools, such as Mean Inequalities. We lay out in detail the benefits of nonroutine solutions to problems, including some positive results according to the practical experience of the authors. Then we give several examples in which the usefulness of this doctrine is demonstrated, including some graphical illustrations. These materials are aimed at teachers, lecturers, and students, and are designed for them to find out about nonstandard teaching and problem-solving methods.