已知函数
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924140810.png)
(
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924187283.png)
为实数).
(Ⅰ)当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924218367.png)
时,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924234447.png)
的最小值;
(Ⅱ)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924234447.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924265523.png)
上是单调函数,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924187283.png)
的取值范围.
(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924296746.png)
(Ⅱ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924312941.png)
第一问中由题意可知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924327391.png)
. ∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924218367.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924359692.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924374679.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924405484.png)
.
当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924421435.png)
时,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924437557.png)
; 当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924468360.png)
时,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924483571.png)
. 故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924296746.png)
.
第二问
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232159245301064.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924405484.png)
.
当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924686370.png)
时,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924702685.png)
,在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924265523.png)
上有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924483571.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924234447.png)
递增,符合题意;
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924842703.png)
,则
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924889740.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924405484.png)
,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924936542.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924952544.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924265523.png)
上恒成立.转化后解决最值即可。
解:(Ⅰ) 由题意可知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924327391.png)
. ∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924218367.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924359692.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924374679.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924405484.png)
.
当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924421435.png)
时,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924437557.png)
; 当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924468360.png)
时,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924483571.png)
. 故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924296746.png)
.
(Ⅱ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232159245301064.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924405484.png)
.
当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924686370.png)
时,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924702685.png)
,在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924265523.png)
上有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924483571.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924234447.png)
递增,符合题意;
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924842703.png)
,则
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924889740.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924405484.png)
,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924936542.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924952544.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924265523.png)
上恒成立.∵二次函数
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924842703.png)
的对称轴为
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215926480526.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215926590573.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215926605862.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232159266521212.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215926683792.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232159267141056.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215926746777.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215926761957.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215926777445.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215926792481.png)
. 综上
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823215924312941.png)
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