【题文】设函数.(1)若求的单调区间及的最小值;(2)若,求的单调区间;(3)试比较与的大小.其中,并证明你的结论.
题型:难度:来源:
【题文】设函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232345-25761.png)
.
(1)若
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-47338.png)
求
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
的单调区间及
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
的最小值;
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-17147.png)
,求
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
的单调区间;
(3)试比较
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232347-55011.png)
与
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232347-18491.png)
的大小.其中
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232347-29041.png)
,并证明你的结论.
答案
【答案】(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-30744.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
的增区间为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-60396.png)
,减区间为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-63806.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-78464.png)
;(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232349-31839.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
的递增区间是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232349-39175.png)
,递减区间是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232349-67623.png)
;当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232350-53210.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
的递增区间是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-60396.png)
,递减区间是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-63806.png)
;(3)由(1)可知,当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232350-68382.png)
时,有
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232351-99847.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232351-91921.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232351-75034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232351-52803.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232352-62237.png)
.
解析
【解析】
试题分析:(1)先求出导函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232352-57845.png)
,解不等式
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232352-86413.png)
和
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232353-58759.png)
,判断函数的单调性即可;
(2)先求出函数的定义域,然后求出函数的导函数,从导函数的二次项系数的正负;根据导函数根的大小,进行分类讨论;最后判断出导函数的符号;利用函数的单调性与导函数符号的关系求出单调性.
(3)将比较所给的两个式子的大小关系,关键是要根据第(1)小问的结论适当的赋特值,建立不等关系:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232351-91921.png)
.然后根据该不等放缩求和即可得出两者的大小关系.
试题解析:(1)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232353-74446.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232353-69783.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232354-57821.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-60396.png)
上是递增的.
当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232354-42592.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232354-57821.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-63806.png)
上是递减的.
故当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-30744.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
的增区间为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-60396.png)
,减区间为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-63806.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-78464.png)
.
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232349-31839.png)
,当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232355-29095.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232355-58353.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232355-81319.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232349-39175.png)
上是递增的;
当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232356-71686.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232356-65501.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232354-57821.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232349-67623.png)
上是递减的.
若
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232350-53210.png)
,当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232355-29095.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232357-80177.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-60396.png)
上是递增的,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232357-84542.png)
上是递减的;
当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232356-71686.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232356-65501.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232356-36331.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232354-57821.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232357-57075.png)
上是递减的,而
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232358-40574.png)
处有意义; 则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-60396.png)
上是递增的,在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-63806.png)
上是递减的.
综上: 当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232349-31839.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
的递增区间是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232349-39175.png)
,递减区间是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232349-67623.png)
;当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232350-53210.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232346-52591.png)
的递增区间是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-60396.png)
,递减区间是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232348-63806.png)
.
(3)由(1)可知,当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232350-68382.png)
时,有
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232351-99847.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232351-91921.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232358-48349.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232358-35539.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232352-62237.png)
.
考点:利用导数求闭区间上函数的最值;利用导数研究函数的单调性.
举一反三
【题文】函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232319-16472.png)
的单调递增区间是( )
A.![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232320-80479.png) | B.(0,3) | C.(1,4) | D.![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232320-69390.png) |
【题文】函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232311-43960.png)
的定义域为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232311-68294.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232311-44578.png)
,对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232312-85424.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232312-27616.png)
,则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232312-50155.png)
的解集为( )
【题文】若函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232302-74958.png)
(x)=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232303-93765.png)
,则该函数在(-∞,+∞)上是( ).
A.单调递减无最小值 | B.单调递减有最小值 |
C.单调递增无最大值 | D.单调递增有最大值 |
【题文】已知偶函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232229-36286.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232229-18428.png)
单调递减,则满足
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232229-24511.png)
的
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232229-33492.png)
的取值范围是( )
【题文】(1)用函数单调性定义证明:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232218-44527.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232218-25745.png)
上是减函数;
(2)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325232218-34552.png)
的值域.
最新试题
热门考点