专题一第6讲母题突破3　零点问题_2.2025数学总复习_赠品通用版（老高考）复习资料_二轮复习_2023年高考数学二轮复习讲义+课件（全国版文科）_学生版_学生用书Word版文档_209

母题突破 3 零点问题
母题 (2022·全国乙卷)已知函数f(x)＝ax－－(a＋1)ln x.
(1)当a＝0时，求f(x)的最大值；
(2)若f(x)恰有一个零点，求a的取值范围．
思路分析
❶fx的单调性
↓
fx的最值
❷求f′x
↓
分类讨论fx的单调性
↓
利用单调性、零点存在定理判断零点个数
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[子题1] (2021·全国甲卷改编)已知a>0且a≠1，函数f(x)＝(x>0)，若曲线y＝f(x)与直线
y＝1有且仅有两个交点，求a的取值范围．
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[子题2] 已知函数f(x)＝＋cos x，求证：当x∈(π，＋∞)时，f(x)有且仅有1个零点．
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规律方法 (1)三步求解函数零点(方程根)的个数问题
第一步：将问题转化为函数的零点问题，进而转化为函数的图象与x轴(或直线y＝k)在该区
间上的交点问题；第二步：利用导数研究该函数在该区间上的单调性、极值(最值)、端点值等性质；
第三步：结合图象求解．
(2)已知零点求参数的取值范围：①结合图象与单调性，分析函数的极值点；②依据零点确
定极值的范围；③对于参数选择恰当的分类标准进行讨论．
1．(2022·河南六市联考)已知函数f(x)＝ex－ax＋2a，a∈R.
(1)讨论函数f(x)的单调性；
(2)讨论函数f(x)的零点个数．
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2．已知函数f(x)＝ln x－x.
(1)求证：f(x)≤－1；
(2)若函数h(x)＝af(x)＋(a∈R)无零点，求a的取值范围．
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