名校
解题方法
1 . 证明:
(1)若
,求证:
;
(2)若
,求证:
.
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aae41b4ea7f44a8699f108def4a22ac9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73c79de030dea51c5e80e233b44788de.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21d2957f2fb09a6caf84dc5a0a8dfead.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5673a0c44c18320db28445ac653a0acc.png)
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解题方法
2 . 阅读:序数属性是自然数的基本属性之一,它反映了记数的顺序性,回答了“第几个”的问题.在教材中有如下顺序公理:①如果
,那么
;②如果
,那么
.
(1)请运用上述公理①②证明:“如果
,那么
.”
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbe96cc58c73271a157f908b4261620a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/069390dd908ff203327958117a226593.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9dac0a497d02926a23678d5dc6bcf79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b34c5832f3fe28f48a924854cb8814ba.png)
(1)请运用上述公理①②证明:“如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a6c4658978e20d4074a1099de1e15a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27f50ce6b511e6b928796e048fc7fa5c.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af26bd7fd83da5267ed64b3f22ad59a0.png)
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名校
解题方法
3 . 已知
均为正实数.
(1)设
,
,求证:
;
(2)若
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/333de134fb62d12d1b62f59bab55fbfb.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b74110bc818c2f5a53d63451c5251eb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/436a2732e9c9d5ce401c448cd9de80e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f663a586008ecff29abc4cba5948830.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0d957381a6902b4d7192f13043aa6a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/660ca2c4e0dc6e567c74066ea95aaeb6.png)
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2022-10-19更新
|
268次组卷
|
2卷引用:江苏省徐州市沛县歌风中学2022-2023学年高一上学期阶段性检测(一)数学试题
4 . (1)证明:若
,
,则
.
(2)利用基本不等式证明:已知
都是正数,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70410f095a6d5b4b66ece2ad7bf1e461.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d5556dd86322752a457b3a6ba979c80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cd8c11384de6e399d7cff57f7824b69.png)
(2)利用基本不等式证明:已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb750904ec9f5877dac7638e45e45936.png)
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名校
解题方法
5 . 证明下列不等式
(1)求证:
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fbbe8678e8336495fe1383b2178ecd8.png)
(2)已知
都是正数,求证:
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f70e0db0174a2c05b28fb6d0c2508778.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fbbe8678e8336495fe1383b2178ecd8.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a14c388e1e2e5a2ff1ccf6caffbee0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f14305fe9ecbe4aa54341b2c7574db.png)
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21-22高一·湖南·课后作业
解题方法
6 . 证明不等式:
(1)若
,
,
,
都是正数,求证:
;
(2)若
,
,
是非负实数,则
;
(3)若
,
是非负实数,则
;
(4)若
,
,则
.
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](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/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2e7387a3fbab6508695365955f55258.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](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/b2d92a6b95fdfdedb405447340293bdc.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/727ff3ac24b506706045956c16336f94.png)
(4)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19339e3904e9541ff26b30ae5f1242b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4f58b9bc974b789928f6490acb43fb3.png)
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2019高三·江苏·专题练习
7 . 利用基本不等式证明:已知
都是正数,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb750904ec9f5877dac7638e45e45936.png)
您最近一年使用:0次
2021-08-31更新
|
2386次组卷
|
15卷引用:专题7.3 基本不等式及其应用(讲)-江苏版《2020年高考一轮复习讲练测》
(已下线)专题7.3 基本不等式及其应用(讲)-江苏版《2020年高考一轮复习讲练测》(已下线)专题7.4 基本不等式及其应用(讲)-浙江版《2020年高考一轮复习讲练测》(已下线)专题2.2 基本不等式及其应用(讲)-2021年新高考数学一轮复习讲练测(已下线)专题2.2 基本不等式及其应用(精讲)-2021年新高考数学一轮复习学与练【新教材精创】3.2.1+基本不等式的证明+学案-苏教版高中数学必修第一册【新教材精创】3.2.1+基本不等式的证明+教学设计-苏教版高中数学必修第一册陕西省咸阳市武功县2020-2021学年高二上学期期中数学试题陕西省汉中市部分高中2020-2021学年高二上学期期中数学试题(已下线)第02讲 基本不等式(教师版)-【帮课堂】2021-2022学年高一数学同步精品讲义(人教A版2019必修第一册)(已下线)专题2.2 基本不等式及其应用(讲)- 2022年高考数学一轮复习讲练测(新教材新高考)2.1.2基本不等式(已下线)第07讲 基本不等式-【暑假自学课】(人教A版2019必修第一册)(已下线)2.2 基本不等式(第1课时)(导学案)-【上好课】(已下线)2.2 基本不等式(第1课时)(分层练习)-【上好课】(已下线)第04讲 基本不等式及其应用(十大题型)(讲义)
20-21高一上·全国·课后作业
名校
8 . 已知
,满足
.
(1)求证:
;
(2)现推广:把
的分子改为另一个大于1的正整数
,使
对任意
恒成立,试写出一个
,并证明之.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c0e9d1ad9561d693958756ee8398218.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce613eaa5df46a50174085ef5d1087fb.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3aec1994a01be9e9335a62177131ee4.png)
(2)现推广:把
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b32df45c5ee591bb2b763deacb26110c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942932aac23ed64c833aacaae02e66bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce613eaa5df46a50174085ef5d1087fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
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2021-04-18更新
|
303次组卷
|
4卷引用:3.2 基本不等式(2)应用与难点(课堂培优)-2021-2022学年高一数学课后培优练(苏教版2019必修第一册)
(已下线)3.2 基本不等式(2)应用与难点(课堂培优)-2021-2022学年高一数学课后培优练(苏教版2019必修第一册)(已下线)3.2.2 基本不等式的应用(练习)-2020-2021学年上学期高一数学同步精品课堂(新教材苏教版必修第一册)江西省抚州市黎川县第一中学2020-2021学年高一下学期期末数学(理)试题(已下线)2.2 (分层练)基本不等式-2021-2022学年高中数学必修第一册课时解读与训练(人教A版2019)
9 . (1)
,
,求证:
(用比较法证明)
(2)除了用比较法证明,还可以有如下证法:
∵
,
,
∴
,
当且仅当
时等号成立,
∴
,
学习以上解题过程,尝试解决下列问题:
1)证明:若
,
,
,则
并指出等号成立的条件.
2)试将上述不等式推广到
个正数
、
…,
、
的情形,并证明.
![](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/ead56bb8f5e7a72e9f8640e795caf68d.png)
(2)除了用比较法证明,还可以有如下证法:
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61ecd584735f1b25c61dbdfb52a0ad7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc6f681a9d3bca1f9c80bee0ffafd55a.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fed66eab193c5df3c72644b8fb3b0ec9.png)
当且仅当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f22fec5a381ae8aca93d876e54c79de.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a79098d29641dc08f82f4a5a32c117ed.png)
学习以上解题过程,尝试解决下列问题:
1)证明:若
![](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/8cec12441802f71e803efaf2c62ee588.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfbefc06b3b4e54a6a1690e870efc69b.png)
2)试将上述不等式推广到
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4f27f84764f1cca89ce3d93fc1cf603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7230de53663c75658c58bbf206a0085.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
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10 . 已知数列
和
满足:
.
(1)设
求
的值;
(2)设
求数列
的通项公式;
(3)设
证明:______.
请从下面①,②两个选项中,任选一个补充到上面问题中,并给出证明.
①
;②
其中
.
注:若两个问题均作答,则按第一个计分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08416257a7c5427d9266e9ee46ee492b.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bb3a4652ccd6c113de9645852973d68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08360bbdf8e90d7d35445ea6e9923658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09a27e634b912cd518c69ff3ffb74db8.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bc35823313a638d6dcf399efaeff9e0.png)
请从下面①,②两个选项中,任选一个补充到上面问题中,并给出证明.
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51206c051804c48be676c6510c63ce3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b09b55a8bc170344adf78ee08ea8892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce671943072553870e3c059e835e980c.png)
注:若两个问题均作答,则按第一个计分.
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