专题一　培优点1　切线放缩--2024年高考数学复习二轮讲义

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这是一份专题一　培优点1　切线放缩--2024年高考数学复习二轮讲义，共3页。试卷主要包含了当x>－1时，ln≤x等内容，欢迎下载使用。

考点一单切线放缩
常见的切线放缩：∀x∈R都有ex≥x＋1.当x>－1时，ln(x＋1)≤x.当x>0时，x>sin x；当x<0时，x例1 (2023·重庆模拟)已知函数f(x)＝sin x－aln(x＋1)．
(1)若a＝1，证明：当x∈[0,1]时，f(x)≥0；
(2)若a＝－1，证明：当x∈[0，＋∞)时，f(x)≤2ex－2.
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规律方法该方法适用于凹函数与凸函数且它们的凹凸性相反的问题(拆成两个函数)，两函数有斜率相同的切线，这是切线放缩的基础，引入一个中间量，分别证明两个不等式成立，然后利用不等式的传递性即可，难点在合理拆分函数，寻找它们斜率相等的切线隔板．
跟踪演练1 (2023·柳州模拟)已知函数f(x)＝ln x＋eq \f(a,x)－2x.
(1)当a>0时，讨论f(x)的单调性；
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(2)证明：ex＋eq \f(a－2x2－2x,x)>f(x)．
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考点二双切线放缩
例2 (2023·福州模拟)已知函数f(x)＝xln x－x.若f(x)＝b有两个实数根x1，x2，且x1________________________________________________________________________
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规律方法含有两个零点的f(x)的解析式(可能含有参数x1，x2)，告知方程f(x)＝b有两个实根，要证明两个实根之差小于(或大于)某个表达式．求解策略是画出f(x)的图象，并求出f(x)在两个零点处(有时候不一定是零点处)的切线方程(有时候不是找切线，而是找过曲线上某两点的直线)，然后严格证明曲线f(x)在切线(或所找直线)的上方或下方，进而对x1，x2作出放大或者缩小，从而实现证明．
跟踪演练2 (2023·山东省实验中学模拟)已知函数f(x)＝(x＋1)(ex－1)，若函数g(x)＝f(x)－m(m>0)有两个零点x1，x2，且x1________________________________________________________________________
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