题目内容
已知
为函数
图象上一点,
为坐标原点,记直线
的斜率
.
(1)若函数
在区间![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049328791.png)
上存在极值,求实数
的取值范围;
(2)当
时,不等式
恒成立,求实数
的取值范围;
(3)求证:
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049141615.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049141522.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049157292.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049172371.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049297579.png)
(1)若函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049313495.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049328791.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049344581.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049359337.png)
(2)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049375358.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049406722.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049422267.png)
(3)求证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240220494371372.png)
(1)
;(2)
;(3)证明过程详见解析.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049469539.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049484519.png)
试题分析:本题主要考查导数的应用、不等式、数列等基础知识,考查思维能力、运算能力和思维的严谨性.第一问,考查求导求极值问题;第二问,是恒成立问题,将第一问的
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049500447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049515824.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049547442.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049547442.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049578654.png)
试题解析:(1)由题意
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049609828.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049625393.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240220496401194.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049656435.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049687622.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049687360.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049703618.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049313495.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049734459.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049749543.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049313495.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049781323.png)
因为函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049313495.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049328791.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049827457.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049843904.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049874617.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049359337.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049905657.png)
(2)由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049406722.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049937858.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049968976.png)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049983791.png)
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050015663.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050015822.png)
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050030379.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050046643.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050077513.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050093490.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050108828.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050139620.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050155491.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050093490.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050186773.png)
所以实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022049422267.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050217541.png)
(3)由(2) 知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050249737.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240220502801322.png)
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050280627.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050311978.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050327815.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050342830.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050358965.png)
将以上
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050389297.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240220504051982.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022050420977.png)
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240220504361360.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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