题目
题型:不详难度:来源:
(1)求a的值;
(2)若对任意的x∈[0,+∞),有f(x)≤kx2成立,求实数k的最小值;
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-55227.png)
解析
试题分析:(1)首先确定函数的定义域,然后求导,利用导数,确定函数的单调区间和极小值,此处,极小值就是最小值,由于最小值为0,可建立关于a的方程,解之即可.(2)通过x=1验证k≤0不满足条件,所以k>0,构造函数g(x)=f(x)-kx2,则g′(x)=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-20169.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-91149.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-55227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-55227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-55227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-32473.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-32473.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-55227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-55227.png)
试题解析:.解:(1)f(x)的定义域为(-a,+∞).
f′(x)=1-
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-89088.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041107-15277.png)
由f′(x)=0,得x=1-a>-a.
当x变化时,f′(x),f(x)的变化情况如下表:
x | (-a,1-a) | 1-a | (1-a,+∞) |
f′(x) | - | 0 | + |
f(x) | | 极小值 | |
故由题意f(1-a)=1-a=0,所以a=1. 5分
(2)当k≤0时,取x=1,有f(1)=1-ln2>0,
故k≤0不合题意. 6分
当k>0时,令g(x)=f(x)-kx2,
即g(x)=x-ln(x+1)-kx2.
g′(x)=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-20169.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-91149.png)
令g′(x)=0,得x1=0,x2=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041107-18221.png)
①当k≥
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-55227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041107-18221.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-55227.png)
②当0<k<
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-55227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041107-18221.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-32473.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-32473.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-32473.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-55227.png)
综上,k的最小值为
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041106-55227.png)
核心考点
试题【已知函数f(x)=x-ln(x+a)的最小值为0,其中a>0.(1)求a的值;(2)若对任意的x∈[0,+∞),有f(x)≤kx2成立,求实数k的最小值;】;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/20191018/20191018041100-88204.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
(1)试确定a,b的值;
(2)讨论函数f(x)的单调区间;
(3)若对任意x>0,不等式f(x)≥﹣2c2恒成立,求c的取值范围.
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041044-78802.png)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041044-91562.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041044-13260.png)
(2)若函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041044-13260.png)
(3)若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041044-66069.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041045-24079.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041045-89170.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041044-13260.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041045-93943.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041045-65404.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041045-53049.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041046-95104.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041045-24079.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041046-84149.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041027-78816.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041027-24025.png)
(Ⅰ)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041027-85816.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041027-89123.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041027-58348.png)
(Ⅱ)如果存在
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041027-25360.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041027-55501.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041028-23345.png)
(Ⅲ)如果对任意的
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041028-55144.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041028-26503.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041028-96541.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041005-84079.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041005-93883.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041005-40514.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041005-90592.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041006-83316.png)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041006-61395.png)
(2)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041005-40514.png)
(3)设斜率为
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041006-88978.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041006-72698.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041006-41692.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041007-30670.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018041007-18275.png)
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