2022年不等式中恒成立问题总结 .pdf
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1、_ 精品资料不等式中恒成立问题在不等式的综合题中，经常会遇到当一个结论对于某一个字母的某一个取值范围内所有值都成立的恒成立问题。恒成立问题的基本类型：类型 1：设)0()(2acbxaxxf，（1）Rxxf在0)(上恒成立00且a；（2）Rxxf在0)(上恒成立00且a。类型 2：设)0()(2acbxaxxf（1）当0a时，,0)(xxf在上恒成立0)(2020)(2fababfab或或，,0)(xxf在上恒成立0)(0)(ff（2）当0a时，,0)(xxf在上恒成立0)(0)(ff,0)(xxf在上恒成立0)(2020)(2fababfab或或类型 3：min)()(xfIxxf恒成立对一
2、切max)()(xfIxxf恒成立对一切。类型 4：)()()()()()()(maxminIxxgxfxgxfIxxgxf的图象的上方或的图象在恒成立对一切恒成立问题的解题的基本思路是：根据已知条件将恒成立问题向基本类型转化，正确选用函数法、最小值法、数形结合等解题方法求解。一、用一次函数的性质_ 精品资料对于一次函数,)(nmxbkxxf有：0)(0)(0)(,0)(0)(0)(nfmfxfnfmfxf恒成立恒成立一利用一元二次函数的判别式对于一元二次函数),0(0)(2Rxacbxaxxf有：（1）Rxxf在0)(上恒成立00且a；（2）Rxxf在0)(上恒成立00且a例 1：若不等式0
3、2)1()1(2xmxm的解集是 R，求 m 的范围。例 2.已知函数)1(lg22axaxy的定义域为R，求实数a的取值范围。二最值法将不等式恒成立问题转化为求函数最值问题的一种处理方法，其一般类型有：1）axf)(恒成立min)(xfa2）axf)(恒成立max)(xfa例 3已知xxxxgaxxxf4042)(,287)(232，当3,3x时，)()(xgxf恒成立，求实数a的取值范围。文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B
4、10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:C
5、B1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8
6、O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI
7、10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L1
8、0I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4
9、 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7
10、O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6_ 精品资料例 4函数),1,2)(2xxaxxxf，若对任意),1x，0)(xf恒成立，求实数a的取值范围。注：本题还可将)(xf变形为2)(xaxxf，讨论其单调性从而求出)(
11、xf最小值。三、分离变量法若所给的不等式能通过恒等变形使参数与主元分离于不等式两端，从而问题转化为求主元函数的最值，进而求出参数范围。这种方法本质也还是求最值，但它思路更清晰，操作性更强。一般地有：1）为参数）aagxf)()(恒成立max)()(xfag2）为参数）aagxf)()(恒成立max)()(xfag实际上，上题就可利用此法解决。略解：022axx在),1x时恒成立，只要xxa22在),1x时恒成立。而易求得二次函数xxxh2)(2在),1上的最大值为3，所以3a。例 5已知函数4,0(,4)(2xxxaxxf时0)(xf恒成立，求实数a的取值范围。注：分离参数后，方向明确，思路清
12、晰能使问题顺利得到解决。四若二次不等式中x的取值范围有限制，则可利用根的分布解决问题。文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J
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14、H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF
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17、CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M
18、8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 Z
19、I10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6_ 精品资料例 6设22)(2mxxxf，当),1x时，mxf)(恒成立，求实数m的取值范围。五、变换主元法处理含参不等式恒成立的某些问题时，若能适时的把主元变量和参数变量进行“换位”思考，往往会使问题降次、简化。例 7对任意 1,1a，不等式024)4(2axax恒成立，求x的取值范围。注：一般地，一次函数)0()(kbkxxf在,上恒有0)(xf的充要条件为0)(0)(ff。六、数形结合法数学家华罗庚曾说过：“数缺形时少直观，形缺数时难入微”，这充分说明了数形结合思想的妙处，在不等
20、式恒成立问题中它同样起着重要作用。我们知道，函数图象和不等式有着密切的联系：1）)()(xgxf函数)(xf图象恒在函数)(xg图象上方；2）)()(xgxf函数)(xf图象恒在函数)(xg图象下上方。x-2-4 y O-4 文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O10B10G4 HF3I9L10I5H3 ZI10J6H8M8O6文档编码:CB1O7O
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