试题分析:(1)先求导,根据题意
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309500592.png)
(2)可将问题转化为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309516669.png)
≥
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309547679.png)
,分别求导令导数大于0、小于0得单调性,用单调性求最值。在解导数大于0或小于0的过程中注意对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309172283.png)
的讨论。
试题解析:(1)解法1:∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309578972.png)
,其定义域为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309594563.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309609943.png)
. ∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309141323.png)
是函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309640513.png)
的极值点,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309500592.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309672506.png)
.
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309141398.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309250444.png)
. 经检验当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309250444.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309141323.png)
是函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309640513.png)
的极值点,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309250444.png)
.、
解法2:∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309578972.png)
,其定义域为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309796560.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309609943.png)
. 令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309812624.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309828730.png)
,整理,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309843641.png)
.
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309843673.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309812624.png)
的两个实根
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309874806.png)
(舍去),
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309890822.png)
,
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309906266.png)
变化时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309640513.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309921546.png)
的变化情况如下表:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/2014082404230993710274.png)
依题意,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309952740.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309968433.png)
,∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309141398.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309250444.png)
.
(2)对任意的
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309188627.png)
都有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309204528.png)
≥
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309219523.png)
成立等价于对任意的
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309188627.png)
都有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309516669.png)
≥
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309547679.png)
.当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309906266.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310171246.png)
[1,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309204264.png)
]时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310202777.png)
.
∴函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309126641.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310233353.png)
上是增函数.∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310233954.png)
.
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240423102641154.png)
,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310280492.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309141398.png)
.
①当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310296437.png)
且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309906266.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310171246.png)
[1,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309204264.png)
]时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240423103581110.png)
,
∴函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309110757.png)
在[1,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309204264.png)
]上是增函数,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240423104201049.png)
.由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310436409.png)
≥
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310452322.png)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309172283.png)
≥
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310467329.png)
,又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310296437.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309172283.png)
不合题意.
②当1≤
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309172283.png)
≤
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309204264.png)
时,
若1≤
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309906266.png)
<
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309172283.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240423106081108.png)
,若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309172283.png)
<
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309906266.png)
≤
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309204264.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240423103581110.png)
.
∴函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309110757.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310701439.png)
上是减函数,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310701455.png)
上是增函数.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310732983.png)
.
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310748362.png)
≥
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310452322.png)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309172283.png)
≥
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310779453.png)
,又1≤
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309172283.png)
≤
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309204264.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310826425.png)
≤
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309172283.png)
≤
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309204264.png)
.
③当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310857387.png)
且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309906266.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310171246.png)
[1,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309204264.png)
]时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240423106081108.png)
,
∴函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309110757.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310233353.png)
上是减函数.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240423109511154.png)
.由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310982510.png)
≥
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310452322.png)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309172283.png)
≥
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310467329.png)
,
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310857387.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042310857387.png)
.
综上所述,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309172283.png)
的取值范围为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824042309266763.png)
.