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对于一些用传统方法难以解决的函数的零点问题,导数是一个非常有效的工具.下面举例说明,供参考.例1已知函数f(x)=(x-2)e~x+a(x-1)~2有两个零点.(1)求a的取值范围;(2)设x_1,x_2是f(x)的两个零点,证明:x_1+x_2<2.(2016年新课标Ⅰ卷)解(1)f′(x)=(x-1)e~x+2a(x-1)=(x-1)(e~x+2a).
Derivatives are a very effective tool for some of the zero-point problems of functions that are difficult to solve with traditional methods. The following examples are given for reference: Example 1 The function f (x) = (x-2) e ~ x + a -1) ~ 2 has two zeros. (1) Find the range of a; (2) Let x_1 and x_2 be two zeros of f (x), prove that: x_1 + x_2 <2. (1) f ’(x) = (x-1) e ~ x + 2a (x-1) = (x-1) (e ~ x + 2a).