试题分析:本题主要考查导数的运算、利用导数求曲线的切线、利用导数判断函数的单调性、利用导数求函数的最值、恒成立问题等基础知识,考查学生的分析问题解决问题的能力、转化能力、计算能力,考查学生的分类讨论思想、函数思想.第一问,对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717808447.png)
求导,将切点的横坐标代入得到切线的斜率,再将切点的横坐标代入到
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717808447.png)
中,得到切点的纵坐标,利用点斜式得到切线的方程;第二问,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717808447.png)
在定义域
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717855474.png)
内是增函数,只需
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717886554.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717855474.png)
恒成立,对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717808447.png)
求导,由于分母恒正,只需分子
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717933672.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717855474.png)
恒成立,设函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717964741.png)
,利用抛物线的性质求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717980555.png)
,令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717996669.png)
即可,解出P的值;第三问,先通过函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718011426.png)
的单调性求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718011426.png)
的值域,通过对P的讨论研究
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717808447.png)
的单调性,求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717808447.png)
的值域,看是否有值大于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718011426.png)
的最小值为2.
(1)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717574404.png)
时,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718183804.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718214686.png)
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718230767.png)
,曲线
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717574429.png)
在点
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717590480.png)
处的切线的斜率为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718292616.png)
.
从而曲线
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717574429.png)
在点
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717590480.png)
处的切线方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718339636.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717762492.png)
.…4分
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240527183861079.png)
.
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717964741.png)
,要使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717574429.png)
在定义域
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717855474.png)
内是增函数,只需
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718448534.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717855474.png)
内恒成立.
由题意
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718479406.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717964741.png)
的图象为开口向上的抛物线,对称轴方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718526741.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718557785.png)
, 只需
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718573533.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718588373.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718588710.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717574429.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717855474.png)
内为增函数,正实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717621308.png)
的取值范围是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717777416.png)
.……9分
(3)∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717652627.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717668358.png)
上是减函数,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718682358.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718807634.png)
;
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718822324.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718838671.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718854663.png)
,
①当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718869409.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717964741.png)
,其图象为开口向下的抛物线,对称轴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718900459.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718916295.png)
轴的左侧,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718932502.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717574429.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718947306.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717668358.png)
内是减函数.
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718978381.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718994553.png)
,因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718947306.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717668358.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052719041533.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052719056777.png)
,
此时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717574429.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718947306.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717668358.png)
内是减函数.
故当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052719119405.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717574429.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717668358.png)
上单调递减
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052719181864.png)
,不合题意;
②当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052719197461.png)
时,由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052719212722.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240527192441284.png)
.
又由(2)知当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052719259353.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717574429.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717668358.png)
上是增函数,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052719509966.png)
,不合题意;
③当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718588373.png)
时,由(2)知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717574429.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717668358.png)
上是增函数,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052719587573.png)
,
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718011426.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717668358.png)
上是减函数,故只需
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052719649742.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052719665476.png)
,
而
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240527196801288.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052718807634.png)
,
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052719727950.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052719743587.png)
,
所以实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717621308.png)
的取值范围是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824052717793781.png)
. 14分