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不等式(或等式)恒成立(或能成立)问题真是常考不衰.其典型的题目表述为:已知含字母x,a的不等式F(x,a)≤(或≥)0在x的给定的取值范围内恒成立,求a的(待定的)取值范围.(“能成立”的情况,请同学自己研究)其基本的解题思想是消元:x在给定的取值范围内变化,但是F(x,a)≤(或≥)0成立不变,我们不能把x的所有可能的值一一代人,并要求F(x,a)≤(或≥)0成立,即不能通过一一代入消去x(因为其数量往往是无限的);我们只能找x的某些特殊的值,使得F(x,a)最大(或小),并要求该最值≤(或≥)0成立,即只能通过求最值消去x,剩下a,然后解不等式,求出范围.其最常用的解题方法是分离参变量:将F(x,a)≤(或≥)0转化成g(a)≤(或≥)f(x)的形式后,条件也就转化成了g(a)≤[f(x)]_(min)(或≥[f(x)]_(max)),接下来只要,先求只含有x的函数式f(x)在x的给定的取值范围内的最小(或大)值(有时最值不存在,但是确界(无限接近的值)存在,那么就要求出确界),再解只含有a的不等式g(a)≤(或≥)最值,即可得a的(待定的)取值范围.为什么要分离参变量呢?因为一般来说,求f(x)(关于x)的最值(常用导数方法,结果一般为定值),比求F(x,a)(关于x)的最值(常要结合图象,结果一般含有a)容易!注意,有时分离参变量很麻烦,甚至做不到,此时也不要直接求F(x,a)(关于x)的最值,应该适当调整F(x,a)≤(或≥)0,得到m(x,a)≤(或≥)h(x,a),使得m(x,a),h(x,a)都是(关于x)比较熟悉、容易处理的函数式.也就是说,要学会观察式子,找到优解!典型的不等式恒成(或能成立)问题之所以重要,是因为有很多与它本质相同,但“已知”或“要求”有些变化的同类(变式)问题.比如,在已知不等式中增加一个给定取值范围的变量,或者增加一个待定取值范围的变量(此时往往要求“关于两个待定取值范围的变量的某个函数式”的取值范围);再如,将不等式换成等式(当然等式是比较“强”的条件,因此经常研究等式能成立问题(即方程有解问题),很少研究等式恒成立问题);还有,不“已知”不等式恒成立,而“已知”函数恒有某定义域、某值域(最值)、某单调性、某对称性等.
The inequality F (x, a) ≤ (or ≥) 0 is known to contain the letter x, a. The inequality F (x, a) (Given the case of “can be established ”, please study with their own students) the basic problem-solving idea is elimination: x in the We assume that F (x, a) ≤ (or ≥) 0 holds indefinitely, and we can not assign all possible values of x one generation per generation and require F (x, a) ≤ (or ≥) 0 holds, that is, x can not be eliminated by one generation (because its number is often infinite); we can only find some special value of x such that F (x, a) is the largest (Or ≥) 0 holds, that is, the range can be obtained only by finding the most significant value of x, leaving a and then solving the inequality. The most common way to solve this problem is to separate the variables: F (a) ≤ [f (x)] _ (min) (a) ≤ (or ≥) 0 into g (a) ≤ (or ≥) f Or ≥ [f (x)] _ (max)), and then only the minimum (or large) value of the function f (x) containing only x in the given range of x Value does not exist, but it does (Infinitely close to the value), then we need to get the exact boundary), and then only contains a inequality g (a) ≤ (or ≥) the most value, you can get a (to be determined) range of values. For the most part, f (x) (for x) (commonly used derivative methods, the result is generally a fixed value) than the maximum value of F (x, a) for x Often combined with the image, the results generally contain a) easy! Note that sometimes it is cumbersome, or even impossible, to separate parameters, at this time do not directly seek F (x, a) (on x) the value of the most appropriate adjustment Let m (x, a) ≤ (or ≥) h (x, a) such that m (x, a), h x) A familiar, easy-to-handle function. That is, learn to look at formulas and find good solutions! The question of whether a constant (or true) canonical inequality is important is because there are many things that are essentially the same, For example, add a variable of a given value range to a known inequality, or add a variable of a range of values to be determined (this Often require “a range of values for a function of the two variables to be determined”); For example, the inequality is replaced by the equation (of course, the equation is more “strong ” conditions, so often to study equations to set up the problem (that is, equations have problems), rarely study the equation is established problem); , Not “known ” inequality holds, and “known ” function always has a certain domain, a range (most value), a monotone, a symmetry and so on.