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§3.5 利用导数研究恒(能)成立问题
课标要求 恒(能)成立问题是高考的常考考点，其中不等式的恒(能)成立问题经常与导数及
其几何意义、函数、方程等相交汇，综合考查分析问题、解决问题的能力，一般作为压轴题
出现，试题难度略大．
题型一 分离参数求参数范围
例1 已知函数f(x)＝ex－ax－1.
(1)当a＝1时，求f(x)的单调区间与极值；
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(2)若f(x)≤x2在(0，＋∞)上有解，求实数a的取值范围．
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跟踪训练1 已知函数f(x)＝ax－ex(a∈R)，g(x)＝.
(1)当a＝1时，求函数f(x)的极值；
(2)若存在x∈(0，＋∞)，使不等式f(x)≤g(x)－ex成立，求实数a的取值范围．
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题型二 等价转化求参数范围
例2 (2023·柳州模拟)已知函数f(x)＝ax－ln x.
(1)讨论函数f(x)的单调性；
(2)若x＝1为函数f(x)的极值点，当x∈[e，＋∞)时，不等式x[f(x)－x＋1]≤m(e－x)恒成立，
求实数m的取值范围．
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跟踪训练2 (2024·咸阳模拟)已知函数f(x)＝ln x＋x＋(a≠0)．
(1)当a＝1时，求f(x)的极值；
(2)若对∀x∈(e－1，e)，f(x)<x＋2，求实数a的取值范围．
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题型三 双变量的恒(能)成立问题
例3 (2023·济南模拟)已知函数f(x)＝－1(其中e为自然对数的底数)，函数g(x)＝x3＋ax2＋1.
(1)求曲线y＝f(x)在点(1，f(1))处的切线方程；
(2)若对∀x，x∈[1，e]，不等式f(x)≤g(x)恒成立，求实数a的取值范围．
1 2 1 2
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思维升华 “双变量”的恒(能)成立问题一定要正确理解其实质，深刻挖掘内含条件，进行
等价变换，常见的等价变换有
对于某一区间I
(1)∀x，x∈I，f(x)>g(x) f(x) >g(x) .
1 2 1 2 min max
(2)∀x
1
∈I
1
，∃x
2
∈I
2
，f(x 1⇔)>g(x
2
) f(x)
min
>g(x)
min
.
(3)∃x
1
∈I
1
，∀x
2
∈I
2
，f(x
1
)>g(x
2
)⇔f(x)
max
>g(x)
max
.
跟踪训练3 已知函数f(x)＝(x∈R)，a为正实数．
⇔
(1)求函数f(x)的单调区间；
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(2)若∀x，x∈[0,4]，不等式|f(x)－f(x)|<1恒成立，求实数a的取值范围．
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