Concavity is a feature of a function. It has geometric and algebraic significance. Concave-convex function has a lot of theoretical research and applications in many aspects such as inequality, functional analysis, optimization theorem, mathematical programming, operational research, cybernetics and so on. With convex sets, this research has formed a special research direction, convex analysis. The background of its appearance is relatively simple, and the importance of concavity can be obtained only from the study of function diagram, but it is worth noting that the mathematical idea of combining graphics and charts shown here is very important.

In this paper, several basic concepts of function concavity are introduced firstly, then several criteria for function concavity are given, and the geometric significance of low-order derivatives is also discussed. Finally, several simple applications of function concavity are introduced. Through the definition of inflection point, the function of concavity in function diagram is demonstrated. Through Zhan Sen inequality, the effective application of concavity in inequality derivation is demonstrated.

Keywords: function, concavity, derivative, inflection point, Zhan Sen inequality.