本题主要考查了函数的基本性质,考查了分类讨论、函数与方程、数形结合数学思想方法,考查转化与化归的能力、逻辑推理能力。
(1)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725451879.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231437254671091.gif)
.
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725482183.gif)
对任意实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725513575.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725529933.gif)
.
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725545304.gif)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725560906.gif)
;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725576297.gif)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231437256071100.gif)
.
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231437256231974.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725638404.gif)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724702287.gif)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724749269.gif)
上为增函数,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724749278.gif)
上为减函数;
(3)由函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724141270.gif)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724453282.gif)
上的单调性可知,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724141270.gif)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724999237.gif)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725279226.gif)
处取得最小值
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725014351.gif)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725404315.gif)
,而在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725030233.gif)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725326226.gif)
处取得最大值
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724609331.gif)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725435374.gif)
.
故有
①
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724780251.gif)
而
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724141270.gif)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724999237.gif)
处取得最小值
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725014351.gif)
,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725030233.gif)
处取得最大值
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724609331.gif)
.
②
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725061241.gif)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724141270.gif)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724999237.gif)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725279226.gif)
处取得最小值
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725295432.gif)
,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725030233.gif)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725326226.gif)
处取得最大值
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725342430.gif)
.
③
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725357301.gif)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143724141270.gif)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725279226.gif)
处取得最小值
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725404315.gif)
,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725326226.gif)
处取得最大值
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143725435374.gif)
.
点评:函数基本性质的考查是高考热点问题之一,从近几年的高考看,函数问题是高考中的重点考查内容之一,分值近40分左右,主要是考查函数解析式、定义域、值域(最值、参数取值范围)、函数的图象、单调性、奇偶性等性质,考查的函数也是常见的二次函数、指数对数函数为主,但会将这几种函数结合起来、将抽象函数与具体函数结合起来的趋势,这种命题的趋势在今后几年内继续保持。