函数凹凸性的等价定义及其证明  

The Equivalent Definitions of Convex/Concave Function and Its Proof

作　　者：张海芳 

机构地区：[1]滇西科技师范学院数理系,云南临沧677000

出　　处：《文山学院学报》2015年第6期63-65,68,共4页Journal of Wenshan University

基　　金：云南省教育厅科研基金项目"不确定复杂动态网络的自适应控制"(2012Z150C)

摘　　要：教科书中,凹凸函数的定义大都从几何意义引出,一般描述为:凸(凹)曲线弧段上任意两点联结而成的弦,总位于曲线弧段的下(上)方;或者,当曲线各点处存在切线时,凸(凹)曲线弧全部位于曲线上各点处切线的下(上)方。前者往往作为定义使用,后者却没有讨论。然而后者是凸(凹)函数的充分必要条件,也可以作为定义使用。所以由后者可引出一个新的定义,并且这些定义是等价的,从而进一步加深和拓宽了对连续函数凹凸性的认识和理解。In the textbook the definition of convex/concave function is mostly from the geometric meaning. Generally can be described as the chord of any two points on the convex/concave curve located in the lower/upper side of the curve or when there is a tangent at each point of the curve, all the convex/concave curve is located at the bottom/top of the tangent of the points on the curve. The former is often used as a definition, but the latter is not discussed. However, the latter is the necessary and sufficient condition of convex/concave function, which can also be used as a definition. So the latter can lead to new definitions and these definitions are equivalent, which further extend the cognition and understanding of convexity/concavity of continuous functions.

关 键 词：凹凸性 凹凸函数 等价性 

分 类 号：O174[理学—数学]