专题1第6讲母题突破2　恒成立问题与有解问题_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（全国版理科）_学生版_学生用书Word版文档_961

母题突破 2 恒成立问题与有解问题
母题 (2020·全国Ⅰ)已知函数f(x)＝ex＋ax2－x.
(1)当a＝1时，讨论f(x)的单调性；
(2)当x≥0时，f(x)≥x3＋1，求a的取值范围．
2思路分析一
❶∀x≥0，fx≥x3＋1
↓
❷分离参数a≥gx
↓
❸a≥gx
max
↓
❹求gx
max
思路分析二
❶∀x≥0，fx≥x3＋1
↓
❷等价变形
↓
❸构造新函数
↓
❹求新函数的最值
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[子题1] 已知函数f(x)＝ex－ax－1.若f(x)≤x2在x∈(0，＋∞)上有解，求实数a的取值范围．
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[子题2] (2022·山东百事联盟联考)已知函数f(x)＝aeax＋a(a>0)，g(x)＝2ln x．若对∀x>0，
f(x)≥g(x)恒成立，求实数a的取值范围．
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规律方法 (1)由不等式恒成立求参数的取值范围问题的策略
①求最值法：将恒成立问题转化为利用导数求函数的最值问题．
②分离参数法：将参数分离出来，进而转化为 a>f(x) 或a<f(x) 的形式，通过导数的应用
max min
求出f(x)的最值，即得参数的范围．
(2)不等式有解问题可类比恒成立问题进行转化，要理解清楚两类问题的差别．
1．(2022·福州模拟)已知函数f(x)＝xln x.
(1)求曲线y＝f(x)在点(1，f(1))处的切线方程；
(2)当x≥1，f(x)≤ax2－a，求a的取值范围．
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2．(2022·榆林模拟)已知函数f(x)＝aex＋1，g(x)＝ln x.
(1)讨论函数h(x)＝g(x)＋的单调性；
(2)若xf(x)<－g(x)＋1，求a的取值范围．
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