2024学生版大二轮数学新高考提高版（京津琼鲁辽粤冀鄂湘渝闽苏浙黑吉晋皖云豫新甘贵赣桂）专题一　第5讲　母题突破1　导数与不等式的证明15

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这是一份2024学生版大二轮数学新高考提高版（京津琼鲁辽粤冀鄂湘渝闽苏浙黑吉晋皖云豫新甘贵赣桂）专题一　第5讲　母题突破1　导数与不等式的证明15，共3页。

母题突破1 导数与不等式的证明
母题 (2023·十堰调研)已知函数f(x)＝(2－x)ex－ax－2.
(1)若f(x)在R上是减函数，求a的取值范围；
(2)当0≤a<1时，求证：f(x)在(0，＋∞)上只有一个零点x0，且x0思路分析
❶f′x≤0恒成立
❷f′xmax≤0求解
❸0❹x0<\f(e,a＋1)，ax0＋x0❺ax0＋x0<2－x0
❻2－x0≤e
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[子题1] (2023·哈师大附中模拟)已知函数f(x)＝ex＋exln x(其中e是自然对数的底数)．
求证：f(x)≥ex2.
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[子题2] 已知函数f(x)＝ln x，g(x)＝ex.证明：f(x)＋eq \f(2,ex)>g(－x)．
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规律方法利用导数证明不等式问题的方法
(1)直接构造函数法：证明不等式f(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．(2023·桂林模拟)已知函数f(x)＝x2－cs x，求证：f(x)＋2－eq \f(x,ex－1)>0.
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2．(2023·南昌模拟)已知函数f(x)＝a(x2－1)－ln x(x>0)．
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