题目
题型:不详难度:来源:
(I)若函数y=f(x)在区间[1,+∞)上是单调递增函数,求实数a的取值范围;
(II)若函数y=f(x)有两个极值点x1,x2且
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050137-31309.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050137-22413.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050137-24371.png)
解析
试题分析:(Ⅰ)利用导数,先对函数进行求导,让
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050137-44674.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050137-47646.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050137-17543.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050138-10645.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050138-80882.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050138-96410.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050138-83160.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050138-17498.png)
试题解析:(Ⅰ)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050137-44674.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050138-14403.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050138-67247.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050138-14403.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050139-35110.png)
经检验, 当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050139-89672.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050139-47619.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050139-47941.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050140-78054.png)
所以满足题意的a的取值范围为
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050137-24371.png)
(Ⅱ)函数的定义域
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050138-10645.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050137-47646.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050137-17543.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050138-10645.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050138-80882.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050138-96410.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050138-83160.png)
法一:
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050140-21383.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050140-49003.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050140-47802.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050140-62357.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050141-34809.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050141-85642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050141-46175.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050141-16483.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050142-70337.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050142-58135.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050142-55996.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050142-25732.png)
![]() | ![]() | ![]() | ![]() |
![]() | - | 0 | + |
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050143-75792.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050144-31774.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050144-22626.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050144-56675.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050144-28184.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050145-95787.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050145-64173.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050145-32588.png)
法二:6分段后面还有如下证法,可以参照酌情给分.
【证法2】
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050145-63034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050145-89284.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050146-22260.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050146-10654.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050146-37225.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050146-90314.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050147-22037.png)
先证
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050147-76228.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050147-38117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050146-37225.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050147-74394.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050147-11037.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050148-23760.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050148-20570.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050148-17521.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050148-84098.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050147-38117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050147-76228.png)
再证
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050149-68705.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050149-63517.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050149-99830.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050149-68564.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050149-40823.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050150-37976.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050150-82795.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050150-52804.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050150-13769.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050150-14570.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050151-65431.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050151-60444.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050151-32755.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050151-61219.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050152-24618.png)
综上可得
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050152-55343.png)
核心考点
试题【设函数F(x )=x2+aln(x+1)(I)若函数y=f(x)在区间[1,+∞)上是单调递增函数,求实数a的取值范围;(II)若函数y=f(x)有两个极值点x】;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
已知函数f(x)=ex+ax-1(e为自然对数的底数).
(Ⅰ)当a=1时,求过点(1,f(1))处的切线与坐标轴围成的三角形的面积;
(II)若f(x)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050127-72772.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050119-86736.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050119-62458.png)
(Ⅰ)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050119-62965.png)
(Ⅱ)是否存在实数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050120-11960.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050120-43513.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050120-11960.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050114-27940.png)
A.![]() ![]() | B.![]() ![]() |
C.![]() ![]() | D.![]() ![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050103-89090.png)
(Ⅰ)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050103-36548.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050104-40434.png)
(Ⅱ)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050104-40434.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050104-79251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050044-76567.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050044-70132.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050045-39085.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050045-17032.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050045-52117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050045-39513.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050045-48779.png)
(Ⅰ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050046-26531.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050046-61703.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050046-26531.png)
(Ⅱ)设不等式
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050046-92995.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050046-32944.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050047-85630.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050047-60750.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018050047-49380.png)
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