题目内容
已知函数
,其中常数a > 0.
(1) 当a = 4时,证明函数f(x)在
上是减函数;
(2) 求函数f(x)的最小值.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240203507561096.png)
(1) 当a = 4时,证明函数f(x)在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350771451.png)
(2) 求函数f(x)的最小值.
解:(1) 当
时,
,利用“定义法”证明。
(2)![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240203508181621.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350802372.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350802733.png)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240203508181621.png)
试题分析:
思路分析:(1) 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350802372.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350802733.png)
(2)应用均值定理及“对号函数”的单调性,分
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350865564.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350880487.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350896461.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350896399.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240203508181621.png)
解:(1) 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350802372.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350802733.png)
任取0<x1<x2≤2,则f(x1)–f(x2)=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350958694.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350974870.png)
因为0<x1<x2≤2,所以f(x1)–f(x2)>0,即f(x1)>f(x2)
所以函数f(x)在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350771451.png)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020351005754.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020351021525.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020351036428.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350865564.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350880487.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020351099447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020351114484.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350896461.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350896399.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020351099447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020350771451.png)
所以当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020351208383.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020351099447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020351239401.png)
综上所述:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240203508181621.png)
点评:中档题,本题综合性较强,研究函数的单调性,可以利用导数,也可以利用常见函数的单调性。应用均值定理,要注意“一正,二定,三相等”。
![](http://thumb2018.1010pic.com/images/loading.gif)
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