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对于一般函数的极值点,教学中多借助几何直观,用自然语言给出函数极值点的描述性定义:若函数f(x)图象在点P(x1,f(x1))处从左侧到右侧由“上升”变为“下降”(函数由单调递增变为单调递减),我们就称f(x1)为函数f(x)的一个极大值,x1为函数f(x)的一个极大值点;类似的,若函数f(x)图象在点P(x2,f(x2))处从左侧到右侧由“下降”变为“上升”(函数由单调递减变为单调递增),我们就称f(x2)为函数f(x)的一个极小值,x2为函数f(x)的一个极小值点.该定义给出了判断极值点的充要条件,揭示了一般函数极值点的本质特征:极值点附近左侧与右侧函数单调性相反[1].
For the extremum points of the general function, the teaching usually uses geometric intuition to give the descriptive definition of the extremum point of the function in natural language: if the function f (x) image is at the point P (x1, f (x1) Let’s say that f (x1) is a maximum value of the function f (x), and x1 is the left-to-right value of the function f Similarly, if the function f (x) image changes from left to right at point P (x2, f (x2)) from “falling ” to We say that f (x2) is a minimum value of the function f (x) and x2 is a minimum value point of the function f (x). The definition of “rising” (monotonically decreasing to monotonically increasing) The necessary and sufficient conditions for judging extremum points are given, and the essential features of extremum points of general functions are revealed: the left and right functions near the extremum points are opposite in monotonicity [1].