试题分析:(Ⅰ)首先
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813270384.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240108133011119.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813317496.png)
有零点而
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813036463.png)
无极值点,表明该零点左右
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813317496.png)
同号,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813426393.png)
,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813442648.png)
的
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813442389.png)
由此可得
(Ⅱ)由题意,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813442648.png)
有两不同的正根,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813489571.png)
.
解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813255575.png)
,设
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813442648.png)
的两根为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813535400.png)
,不妨设
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813551409.png)
,因为在区间
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813567748.png)
上,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813582585.png)
,而在区间
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813598500.png)
上,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813613578.png)
,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813629319.png)
是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813036463.png)
的极小值点.因
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813036463.png)
在区间
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813598500.png)
上
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813036463.png)
是减函数,如能证明
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813691814.png)
则更有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813707685.png)
由韦达定理,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813723692.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240108137381558.png)
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813754529.png)
其中
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813769349.png)
设
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813785797.png)
,利用导数容易证明
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813801405.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813816334.png)
时单调递减,而
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813832470.png)
,因此
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813847495.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813036463.png)
的极小值
(Ⅱ)另证:实际上,我们可以用反代的方式证明
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813036463.png)
的极值均小于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813910383.png)
.
由于两个极值点是方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813442648.png)
的两个正根,所以反过来,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813972667.png)
(用
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813972291.png)
表示
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813161283.png)
的关系式与此相同),这样
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240108142061321.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010814222840.png)
,再证明该式小于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010813910383.png)
是容易的(注意
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010814269406.png)
,下略).
点评:对于函数与导数这一综合问题的命制,一般以有理函数与半超越(指数、对数)函数的组合复合且含有参量的函数为背景载体,解题时要注意对数式对函数定义域的隐蔽,这类问题重点考查函数单调性、导数运算、不等式方程的求解等基本知识,注重数学思想的运用