10-11高二下·福建三明·阶段练习
1 . 先阅读下列不等式的证法,再解决后面的问题:
已知
且
,求证![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84c6c22cb280913922a6b95344ad0997.png)
证明:构造函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31dba68d676bf9e77c974fb7cc1da4ee.png)
因为对一切
,恒有
,所以
,从而![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84c6c22cb280913922a6b95344ad0997.png)
(1)若
,且
,请写出上述结论的推广式;
(2)参考上述证法,对你的结论加以证明;
(3)若
,求证
已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8c38216ac0eb2bcaa85fd8c61901f8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8e5dadfc7d9b12bea102cc0ce5bfcc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84c6c22cb280913922a6b95344ad0997.png)
证明:构造函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31dba68d676bf9e77c974fb7cc1da4ee.png)
因为对一切
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22e38c541dec8fce1d26886e5ef7d21f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00148b121ca924133dfc93af99e7cb40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84c6c22cb280913922a6b95344ad0997.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24b2509af4d6b78ffe1fc3b4ca7301c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b4b9eca505fbff0bff1d2724d7155eb.png)
(2)参考上述证法,对你的结论加以证明;
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03858c5f3aef6ff80609692d4e4f6fc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1244e50a8246eb77b35b19007fbb443.png)
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2 . 已知函数
,对任意的
,恒有
.
(1)证明:
.
(2)若对满足题设条件的任意
,
,不等式
恒成立,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5109b591b55b04ab757a7abb47bb8cc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53b16fe21c80d4ecb12b8470f3eaab9b.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4907c161ea6aa9577a7f07a2013416f9.png)
(2)若对满足题设条件的任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe0bad0ea73d5cdf689bfaadfa9834ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
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