已知實(shí)數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027470406.png)
,函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240120274851002.png)
.
(Ⅰ)若函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027501448.png)
有極大值32,求實(shí)數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027517277.png)
的值;
(Ⅱ)若對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027532558.png)
,不等式
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027532665.png)
恒成立,求實(shí)數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027517277.png)
的取值范圍.
(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027579442.png)
(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027579840.png)
試題分析:解:(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240120276101048.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240120276261360.png)
2分
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027641538.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027657865.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027673491.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027688379.png)
4分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240120277041076.png)
有極大值32,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027719527.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027735475.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027751472.png)
時(shí)取得極大值 5分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027766967.png)
6分
(Ⅱ)由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027782927.png)
知:
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027782383.png)
時(shí),函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027797449.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027813493.png)
上是增函數(shù),在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027829461.png)
上是減函數(shù)
此時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240120278441015.png)
7分
又對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027532558.png)
,不等式
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027875670.png)
恒成立
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027891730.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027922493.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027938574.png)
9分
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027953383.png)
時(shí),函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027797449.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027813493.png)
上是減函數(shù),在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027829461.png)
上是增函數(shù)
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012028078643.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012028343509.png)
,
此時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012028375824.png)
11分
又對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027532558.png)
,不等式
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027875670.png)
恒成立
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012028421656.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012028437500.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012028453590.png)
13分
故所求實(shí)數(shù)的取值范圍是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012027579840.png)
14分
點(diǎn)評(píng):主要是考查了導(dǎo)數(shù)在研究函數(shù)中的運(yùn)用,通過(guò)導(dǎo)數(shù)的符號(hào)以及極值來(lái)得到最值,求解參數(shù)的范圍,屬于中檔題。
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