Concavity And Convexity Of Functions

After completing this chapter, students will be able to:Understand the difference between a convex function and a convex set.Derive the Hessian matrix and the bordered Hessian matrix.Use the Hessian matrix to identify concave and convex functions.Use the bordered Hessian matrix to identify quasi-concave and quasi-convex functions.The present chapter focuses on the shape of functions. In doing so, it aims to take the reader one step closer to the study of optimization problems. An objective is to introduce and discuss the concept of a convex set, which is essential as concave or convex functions typically need their domains to be a convex set. The chapter then describes conditions under which a function is concave or convex. For a function of one variable, a straightforward condition for concavity and convexity is provided by the second derivative, whereas for functions of more than one variable the Hessian matrix is employed. Finally, the notion of concavity and convexity is generalized by describing conditions under which a function is quasi-concave and quasi-convex.