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函数与不等式关系密切,尤其是含参数的不等式问题,变量较多.处理这类问题,对思维能力的要求很高,稍不注意,便会引起思维混乱导致半途而废,得不出结果.遇到这类问题时,我们应如何处理呢? 例 1 如果 2x-1>m(x2-1)对任m∈[-2,2]都成立,求x的范围. 分析:解题时易想到,由原不等式解出x,再根据m的范围确定x的范围.可以想象,此法解题过程非常烦琐,很难解出结果.应如何考虑呢?注意到m的范围己确定,转换一下角度,把所给不等式看成m的不等式如何?原不等式变形为m(x2-1)-(2x-1)<0,左边显然是m的一次函数.记作f(m),由题意,f(m)<0对任m∈[-2,-]恒成立,由一次函数性质只需f(-2)<0 f(2)<0即可,这样便可解这个关于x的不等式
Functions and inequalities are closely related, especially inequality problems with parameters, there are many variables. To deal with such problems, the demand for thinking ability is very high, a little carelessness will lead to confusion and lead to half-way and no results. In this type of problem, how should we deal with it? Example 1 If 2x-1>m(x2-1) holds for any m∈[-2,2], find the range of x. Analysis: It is easy to think when solving a problem. The original inequality is solved for x, and then the range of x is determined according to the range of m. It can be imagined that the problem solving process of this method is very tedious and it is difficult to solve the result. How should it be considered? The scope of m has been determined, and the angle is changed. What is the inequality of m given by the inequality? The original inequality is transformed into m(x2-1)-(2x-1)<0, and the left side is obviously a linear function of m. Let’s denote it as f(m). f(m)<0 is valid for any m∈[-2,-] constant, and the property of the linear function is f(-2)<0 f(2)<0, so that we can solve this inequality about x.