Characterization of Quasi-Concave Functions and Its Optimality Conditions In N

Let T be open, nonempty subset of n  and : fT   be a function on T .In general if f is concave, every local maximum of f is also a global maximum and first order conditions are sufficient to identify global of concave or convex optimization problems and they are so strict and quite restrictive as an assumption.In this paper we present characterizations of optimization problems under a weakening of the condition of concave functions, called "quasi-concave" functions.Some of the characterizations of these functions as a generalization of concave functions and proves are discussed in terms of its contour sets (level sets), derivatives and extreme properties of the functions on line segment.It also discuses its optimality conditions