试题分析:(1)函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250482495.png)
的定义域为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250685566.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240052507631017.png)
.
① 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250778369.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250794707.png)
,∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250809414.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250841605.png)
,∴ 函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250482495.png)
单调递增区间为
② 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250887403.png)
时,令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250903607.png)
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250919675.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250934566.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250950508.png)
.
(ⅰ)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250981425.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250997481.png)
时,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251012597.png)
,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251028624.png)
,
∴ 函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250482495.png)
的单调递增区间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250685566.png)
.
(ⅱ)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251090426.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251106467.png)
时,方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250934566.png)
的两个实根分别为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251153761.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251168784.png)
.
若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251184556.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251199566.png)
,此时,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251418688.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251433622.png)
.
∴函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250482495.png)
的单调递增区间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250685566.png)
,若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250575398.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251511584.png)
,此时,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251527621.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251433622.png)
,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251558680.png)
时,
∴函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250482495.png)
的单调递增区间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240052516211026.png)
,单调递减区间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240052506381079.png)
.
综上所述,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250575398.png)
时,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250482495.png)
的单调递增区间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240052506071021.png)
,单调递减区间
为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240052506381079.png)
;当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250653396.png)
时,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250482495.png)
的单调递增区间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250685566.png)
,无单调递减区间.
(2)由(1)得当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250653396.png)
时,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250482495.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250685566.png)
上单调递增,故函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250482495.png)
无极值
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250575398.png)
时,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250482495.png)
的单调递增区间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240052506071021.png)
,单调递减区间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240052506381079.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250482495.png)
有极大值,其值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251917985.png)
,其中
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251168784.png)
.
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251948616.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251979552.png)
, ∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005251995868.png)
.
设函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240052520261032.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005252042849.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005252057837.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250685566.png)
上为增函数,又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005252104533.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005252120568.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005252135256.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005252151359.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005252182510.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005252198645.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005252213313.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005252135256.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005252291417.png)
.
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005252307714.png)
,结合
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250575398.png)
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005252338493.png)
,∴实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250497283.png)
的取值范围为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824005250700505.png)
.
点评:本题考查利用导数研究函数的单调性,利用导数研究函数的极值,突出分类讨论思想与转化思想的渗透与应用,属于难题,第二题把有正的极大值的问题转化为图象开口向下与X轴有两个交点,思路巧妙,学习中值得借鉴.