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关于函数的单调性,课本只给出了一个定义,但用定义直接求出函数的单调区间有一定的局限性,往往避不开复杂的讨论。本文给出若干有关函数单调性的命题,盼能给学生以帮助。为了行文方便,约定y=f(x)在某区间上单调增记作“↑”,单调减记作“↓”。如在[a,b]上单调增记作“[a↑b]”,在区域D上单调减记作“D↓”。其它情况仿上标记。〈一〉奇偶函数的单调性关于原点的对称区间上奇函数有相同的单调性,偶函数有相反的单调性。(读者自证) 利用奇函数与偶函数的这一关系在研究
With regard to the monotonicity of the function, the textbook only gives a definition, but using the definition to directly determine the monotonic range of the function has certain limitations, and often cannot avoid complex discussions. This article gives a number of propositions about the monotonicity of functions and hopes to help students. For the convenience of writing, it is agreed that y=f(x) is monotonically incremented as “↑” in a certain range and monotonically reduced as “↓”. If monotonically added as [a,b] on [a,b], it is reduced monotonously to “D↓” in area D. In other cases, imitate the mark. The monotonicity of the <1> parity function has the same monotonicity with respect to the odd function on the symmetry interval of the origin, and the even function has the opposite monotonicity. (Readers prove that this relationship between odd and even functions is used in research