The concavity is a charateristic feature of a function.It has geometric and algebraic meanings.The concavity and convex(concave) functions has massive applications in inequality,functional analysis,optimization theorem,mathematical planning,operational research,cybernetics and many other mathematical,physical and economical aspects for theoretical research and application.With convex set,this aspect of study has formed a particular research direction,convex analysis.The background of its appearance is relatively simple,which is that merely from function graph studies,the importance of concavity can be drawn,but it is significant to be noted that the mathematical thought of the combination of figure and chart shown here is very important.
This article firstly describes several fundamental concepts of function concavity,secondly provides a few conditions for determining the concavity of a function,where the geometric meaning of low-level derivatives is also discussed,and finally introduces a few simple applications of function concavity.With the definition of inflexion,it displays the function of concavity in a function plot.With Jensen's Inequality,it showcases the effective usage of concavity in inequality deduction.
Key words:function,concavity,derivative,inflexion,Jensen's Inequality