1 . “让式子丢掉次数”—伯努利不等式(Bernoulli’sInequality),又称贝努利不等式,是高等数学分析不等式中最常见的一种不等式,由瑞士数学家雅各布.伯努利提出,是最早使用“积分”和“极坐标”的数学家之一.贝努利不等式表述为:对实数
,在
时,有不等式
成立;在
时,有不等式
成立.
(1)证明:当
,
时,不等式
成立,并指明取等号的条件;
(2)已知
,…,
(
)是大于
的实数(全部同号),证明:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30cdfc52dbd70827de9e15fffe39c321.png)
(3)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bc98a4d9ae0580aa2c1152ffb770d4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c4fb8df3614557f13bdc68378437e90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3d4045366a437d4003259050718e244.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f75f0daa973c8fc183b7d21bafd7e8cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c78998ba5f2665a1753c3fa84751716.png)
(1)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65a40142c84be68ee2918c3a8303388c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bc98a4d9ae0580aa2c1152ffb770d4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5026dc5ead3b5adf0e5f4b3e7c4eca1d.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a1cc5cfec94bc5686b41b043acdc8ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30cdfc52dbd70827de9e15fffe39c321.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6b29215b2a741c01efc27199e6c6925.png)
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2024-05-30更新
|
285次组卷
|
3卷引用:江西省鹰潭市2024届高三第二次模拟考试数学试卷
2 . 如图,直线k过圆O的中心,直线
,垂足为M,直线l上不同的三点A,B,C在圆外,且位于直线k上方,A点离M点最远,C点离M点最近,AP,BQ,CR为圆O的三条切线,P,Q,R为切点,试证:
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/9/894704d3-9f13-4cc7-9bd4-c1b3a1b5400c.png?resizew=228)
(1)l与圆O相交时,
;
(2)l与圆O相离时,
;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a3f2e7ebe1d90ccde6d385185c88c61.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/9/894704d3-9f13-4cc7-9bd4-c1b3a1b5400c.png?resizew=228)
(1)l与圆O相交时,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/038927d5acaccbefff48a43c3c002d33.png)
(2)l与圆O相离时,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/153a0382f05239a3f343b6768a2bd8d6.png)
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3 . 已知整数
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe4f83589ccd37deb53db370e8e631e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04ed2b042eec7bb3407d5ca5ba1220f1.png)
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解题方法
4 . 已知:
,
,
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5be97cd1c7111b654d87d8fbb63b6a84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df25107867e1b2ae592119c1408643ff.png)
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5 . 已知a,b为正数,且
,证明
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2958030ec9d7543dda1f529593a915e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a887cfcc109d7bdaff34f04d589eeffe.png)
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6 . 给定正整数
.求最大的实数
.使得
对任意正实数
恒成立,其中
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/526e037512bd10e65ebd99b73aa26341.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/897b606fdc64a88a0938d3d60c3ea3e9.png)
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7 . 设
满足
,则
的最大值是___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a521891098b625f372ff648d110afe1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30ecaa22eff643cdb7b78ad9eca37307.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1c9aed10b511457cd2c49f7de978c19.png)
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8 . 已知
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1867027eb8b757f45fe3615b4f5568d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f37dd18e733e38302305afc8099879.png)
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9 . 求具有下述性质的最大整数m:对全体正整数的任意一个排列
,总存在正整数
,使得:
构成公差为奇数的等差数列.(可以认为:两项也是等差的)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8748938373768d7172dc2a8d43ee2d4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0844d2b5218031f4a67807468b02653c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82079a5446d448fb1bea730b968d7e02.png)
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10 . 设
均为正数,证明:
(1)若
,则
;
(2)若
,则
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6aa7413d61d5f084dff82223388b5c5.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bde47288bc17f24c9a05ced2e48cb5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bddaa8662b7e7df9e093b924ea7aaea.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23067dc8ae6048baaf5507cbc5b76172.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6faf3b389d760578d8c1c3934c56c969.png)
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