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第 6 讲 导数的综合应用
[考情分析] 1.利用导数研究函数的单调性与极值(最值)是高考的常见题型，而导数与函数、
不等式、方程、数列等的交汇命题是高考的热点和难点.2.多以解答题的形式压轴出现，难度
较大．
母题突破 1 导数与不等式的证明
母题 (2022·安康模拟)已知函数f(x)＝ln x－x－1，g(x)＝xex－2x－m，若m≤1，证明：当
x>0时，f(x)0时，g(x)<1.
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[子题2] (2022·九江模拟)已知函数f(x)＝ln x.
求证：e2f(x)g(x)(或 f(x)0(或 f(x)－
g(x)<0)，进而构造辅助函数h(x)＝f(x)－g(x)．
(2)适当放缩构造法：一是根据已知条件适当放缩；二是利用常见放缩结论．
(3)构造“形似”函数，稍作变形再构造，对原不等式同结构变形，根据相似结构构造辅助函
数．
1．已知函数f(x)＝.
(1)求f(x)在R上的极值；
(2)求证：0.
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