题目
题型:不详难度:来源:
(1)当a=-1时,求f(x)的最大值;
(2)当a=-1时,试推断方程|f(x)|=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031939-10030.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031939-83750.png)
答案
解析
f′(x)=-1+
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031940-21433.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031940-49427.png)
当0<x<1时,f′(x)>0;
当x>1时,f′(x)<0.
∴f(x)在区间(0,1)上是增函数,在区间(1,+∞)上是减函数.
f(x)max=f(1)=-1.
(2)由(1)知当a=-1时,
f(x)max=f(1)=-1,∴|f(x)|≥1.
令g(x)=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031939-10030.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031939-83750.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031940-33675.png)
令g′(x)=0,得x=e,
当0<x<e时,g′(x)>0,g(x)在区间(0,e)上单调递增;
当x>e时,g′(x)<0,g(x)在区间(e,+∞)上单调递减.
∴g(x)max=g(e)=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031940-92787.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031939-83750.png)
∴g(x)<1.∴|f(x)|>g(x)恒成立,
即|f(x)|>
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031939-10030.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031939-83750.png)
∴方程|f(x)|=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031939-10030.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031939-83750.png)
核心考点
试题【已知函数f(x)=ax+ln x,其中a为常数,e为自然对数的底数.(1)当a=-1时,求f(x)的最大值;(2)当a=-1时,试推断方程|f(x)|=+是否有】;主要考察你对常见函数的导数等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031934-15910.png)
(1)当a=1时,求曲线y=f(x)在点(2,f(2))处的切线方程;
(2)求f(x)在区间(0,e]上的最小值.
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031918-25678.png)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031918-92838.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031919-87335.png)
(2)记函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031919-21714.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031919-23153.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031919-74188.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031919-23153.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031919-23153.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031920-47020.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031920-54261.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031919-23153.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031920-16658.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031920-99519.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031920-10396.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031919-87335.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031910-65079.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031910-15713.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031910-61549.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031910-90650.png)
(1)若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031911-60689.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031911-76072.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031911-81752.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031911-92021.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031904-40150.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031904-22068.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031904-50754.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031852-33989.png)
(1)若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031852-32772.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031852-43819.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031852-91038.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031853-32251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031853-22489.png)
(3)设
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031853-18635.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018031853-75061.png)
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