关于函数曲线凸向的一个推测及有关结论

A Conjecture on the Convexity of Functions and Some Conclusions

从数学教学研究的角度由微观经济学中短期边际成本曲线的凸向问题引申出在有限开区间可导(且严格递增)的函数当自变量从左侧趋近于某一点时为正无穷大的情况下其函数曲线不可能为上凸的推测并给出解析,构成了一个有意义的启发式、探究式预设教学案例.另给出了一个反映可导函数当自变量增大时,函数增量递增(递减)与导数递增(递减)关系的命题.

From the perspective of mathematical teaching research,the convexity problem of short-term marginal cost curves in microeconomics leads to a conjecture that if a function is differentiable(and strictly increasing)in a finite open interval,and its limit is positive infinity when the variable approaches a certain point from the left,then the function cannot be convex.This constitutes a predetermined meaningful heuristic and exploratory teaching case.In addition,a proposition reflecting the relationship between increasing(decreasing)function and increasing(decreasing)derivative function is given.