专题一第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（全国版文科）_学生版

母题突破 2 恒成立问题与有解问题
母题 已知函数f(x)＝axln x，当x≥1时，f(x)≤x3恒成立，求实数a的最大值．
思路分析一
❶x≥1，fx≤x3
↓
❷aln x－x2≤0
↓
❸(aln x－x2) ≤0
max
↓
❹求φx＝aln x－x2x≥1的最大值
思路分析二
❶x≥1，fx≤x3
↓
❷分离参数得a≤
↓
❸a≤
↓
❹求gx＝x>1的最小值)
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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．已知函数f(x)＝ax－ex.
(1)讨论f(x)的单调性；
(2)若对∀x∈(1，＋∞)，有f(x)<a(ln x＋1)－ex.求实数a的取值范围．
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2．(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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