题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032843-94062.png)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032843-83977.png)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032843-70459.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032843-47213.png)
(ⅰ)证明:当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032843-85645.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032844-27148.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032844-94318.png)
(ⅱ)若当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032844-87499.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032844-27148.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032844-94318.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032844-16167.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032845-81163.png)
解析
试题分析:(1)先求
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032843-94062.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032845-40176.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032845-40176.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032843-83977.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032843-83977.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032843-85645.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032844-27148.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032844-94318.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032845-53066.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032845-95921.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032844-87499.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032844-27148.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032844-94318.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032845-96818.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032846-73632.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032845-95921.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032844-16167.png)
试题解析:(1)求导数,得f ′(x)=ex-1.
令f ′(x)=0,解得x=0.
当x<0时,f ′(x)<0,∴f(x)在(-∞,0)上是减函数;
当x>0时,f ′(x)>0,∴f(x)在(0,+∞)上是增函数.
故f(x)在x=0处取得最小值f(0)=0. 4分
(2)设h(x)=f(x)-g(x)=ex-1-x-ax2,则h′(x)=ex-1-2ax.[
(ⅰ)当a=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032846-99252.png)
h(x)=ex-1-x-
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032846-99252.png)
∵h(0)=1-1=0,∴h(x)存在零点x=0.
由(1),知ex≥1+x,∴h′(x)=ex-1-x≥0,
∴h(x)在R上是增函数,∴h(x)在R上有唯一的零点.
故当a=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032846-99252.png)
(ⅱ)当x>0时,y=f(x)的图象恒在y=g(x)的图象的上方
⇔当x>0时,f(x)>g(x),即h(x)=ex-1-x-ax2>0恒成立.
由(1),知ex≥1+x(当且仅当x=0时等号成立),
故当x>0时,ex>1+x.
h′(x)=ex-1-2ax>1+x-1-2ax=(1-2a)x,
从而当1-2a≥0,即a≤
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032846-99252.png)
∴h(x)在(0,+∞)上是增函数,又h(0)=0,
于是当x>0时,h(x)>0.
由ex>1+x(x≠0),可得e-x>1-x(x≠0),
从而当a>
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032846-99252.png)
故当x∈(0,ln2a)时,h′(x)<0,
此时h(x)在(0,ln2a)上是减函数,又h(0)=0,
于是当x∈(0,ln2a)时,h(x)<0.
综上可知,实数a的取值范围为(-∞,
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032846-99252.png)
核心考点
试题【已知函数(1)求的最小值;(2)设,.(ⅰ)证明:当时,的图象与的图象有唯一的公共点;(ⅱ)若当时,的图象恒在的图象的上方,求实数的取值范围.】;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/20191018/20191018032839-23112.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032839-22583.png)
A.![]() | B.5 | C.4 | D.3 |
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032826-19170.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032827-66458.png)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032827-77969.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032826-19170.png)
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032827-79412.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032827-31029.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032827-75394.png)
(1)求函数f(x)的单调区间;
(2)求f(x)在区间[t,t+2](t>0)上的最小值;
(3)对一切的x∈(0,+∞),2f(x)<g′(x)+2恒成立,求实数a的取值范围.
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018032809-62398.png)
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