Funções convexas e côncavas: aplicação do truque da reta tangente e da desigualdade de Jensen em problemas de olimpíadas de matemática

Abstract

Mathematical competitions serve important educational and social purposes by fos- tering interest in mathematics, developing logical reasoning, and identifying talented students regardless of their social background. Among the recurring topics in such competitions, mathematical inequalities stand out, often requiring more advanced te- chniques for their resolution. This work presents two effective strategies for tackling these problems: Jensens inequality and the tangent line trick. We begin with a brief re- view of derivatives, followed by the introduction of key properties of convex and concave functions, which provide the theoretical foundation for the methods explored. Additio- nal tools are also discussed, such as homogeneous inequalities, the CauchySchwarz inequality, and the relationships among the harmonic, geometric, arithmetic, and qua- dratic means. These concepts are then applied to the resolution of selected problems from recent mathematical olympiads, illustrating the practical utility of the techniques studied.