笔者在潜心研究近年高考数学试题中发现,大量高考题命制的几何背景是曲线与切线动(或静)态的上下位置关系,进而应用函数的凸凹性可获得思路自然、过程简明的图解.这类高考题每年必考、倍受青睐,独具匠心、不断创新.命题形式不仅有含(或不含)参不等式恒成立与存在性问题,而且有曲线切线的条数、曲线公共点与函数零点或极值点的个数等问题.现整理如下,供大家参考.一、预备知识在[a,b]上,f″(x)=[f′(x)]′存在. In my meticulous study of college entrance examination mathematics test questions in recent years, I found that the geometric background of a large number of college entrance examination questions is the relationship between the curve and the tangent (or static) state of the upper and lower position, and then the convexity and concavity of the function can be used to obtain a natural and process diagram. This type of college entrance examination questions must be taken every year, be favored, ingenious and innovative. The propositional form not only contains (or does not contain) the inequality constant establishment and existence problems, but also the number of curve tangents, curve common points and function zero Problems such as the number of extreme points, etc., are now collated for your reference. I. Preliminaries In [a,b], f′′(x)=[f′(x)]′ exists.