题目内容
(本题满分13分)设函数
,且
,
,求证:(1)
且
;
(2)函数
在区间
内至少有一个零点;
(3)设
是函数
的两个零点,则
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425695814.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425711677.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425726631.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425742416.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425758669.png)
(2)函数
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425773495.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425789501.png)
(3)设
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425789424.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425773495.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425836927.png)
(1)根据
,求出
,再根据
即可得证;(2)先求出
和
,根据零点存在定理分
和
讨论即可得证;
(3)利用韦达定理和第(1)问的结论即可得证.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425711677.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425867671.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425726631.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425898430.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425914460.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425945368.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425960366.png)
(3)利用韦达定理和第(1)问的结论即可得证.
试题分析:(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425976928.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425992686.png)
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426007407.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425726631.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426038543.png)
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426054877.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426054428.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426070693.png)
(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240024260851167.png)
①当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425945368.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426116616.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426132772.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426148195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425773495.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426179459.png)
②当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425960366.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426132772.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426210726.png)
函数
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425773495.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426241479.png)
综上所述:函数
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425773495.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425789501.png)
(3)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426288235.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425789424.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425773495.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426148195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240024263501031.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426148195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240024263662262.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426382958.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426288235.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425758669.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002426428191.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002425836927.png)
点评:证明此类问题时,要充分利用不等式的性质和题设条件,尽量每一步都做到言之有据.
![](http://thumb.zyjl.cn/images/loading.gif)
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