1 . 不等式的基本性质:(1)传递性:__________ .
(2)可加性:___________ .
(3)可积性:①___________ ;②___________ .
(4)同向可加性:___________ ;异向可减性:___________ .
(5)同向正数可乘性___________ ;异向异号可乘性:___________ ;异向正数可除性:___________ .
(6)乘方法则:___________ (
,
).
(7)开方法则:___________ (
,
).
(8)倒数法则:___________ ;___________ .
(2)可加性:
(3)可积性:①
(4)同向可加性:
(5)同向正数可乘性
(6)乘方法则:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/209559aca6bf32705588b6a40e0b7320.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
(7)开方法则:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/209559aca6bf32705588b6a40e0b7320.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
(8)倒数法则:
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名校
解题方法
2 . 柯西是一位伟大的法国数学家,许多数学定理和结论都以他的名字命名,柯西不等式就是其中之一,它在数学的众多分支中有精彩应用,柯西不等式的一般形式为:设
,则
当且仅当
或存在一个数
,使得
时,等号成立.
(1)请你写出柯西不等式的二元形式;
(2)设P是棱长为
的正四面体
内的任意一点,点
到四个面的距离分别为
、
、
、
,求
的最小值;
(3)已知无穷正数数列
满足:①存在
,使得
;②对任意正整数
,均有
.求证:对任意
,
,恒有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81a8a1b208f491296432e9e6bf0e91c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0653d6a0e8778ad47b06d5f6b88cffa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/419c991c4022ef12d4801e119018b587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f31a068fb311eff550b3088a212fb2f0.png)
(1)请你写出柯西不等式的二元形式;
(2)设P是棱长为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5edf900c810371fb21297c15f86d8743.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b31ac1def558351e2e3ed1235c570530.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/342d0252c1b2f7d2a84b5c985d19d547.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d31659f106fba3c9750661eb0e3c3eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dde93376f5d29f8f7d501122759b0ab.png)
(3)已知无穷正数数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c24ecf9e59082e563372b12981d03fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ee33826e02eda7aa6221649355a5709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9db6b0bf3d360830fff618193c595b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a33ac34aa03dc7f0a5faad6dc664ec6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cca1d86c9f078347773f700fee49d1d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d191d6de821fbb06a51b5a20112db6de.png)
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2024-05-20更新
|
525次组卷
|
4卷引用:河北省邯郸市2024届高三下学期高考保温数学试题
名校
3 . 已知函数
,函数
.
(1)若
,求
的值域;
(2)若
:
(ⅰ)解关于
的不等式:
;
(ⅱ)设
,若实数
满足
,比较
与
的大小,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b25d5c7479bf62d5c9fc71bf46b56866.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd6cc4bdc89f1475cd1b3e21808ff6a3.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd876a2ed79c64bacc3e64b8ee92735e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b77589ec03475a3a653d684f6f23b467.png)
(ⅰ)解关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa18838a13fda4e45612c32cdf98b71.png)
(ⅱ)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/360ff131c51a4ef6745538c18cec92c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/604be3e10ddd5e520c921a8e5ab923e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a99a77bd57c52838c723803db147e17d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ca8b26c3ad6d892590290a2304126bd.png)
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2024·全国·模拟预测
4 . 设
,
,…,
(
),
,
,…,
(
)为两组正实数,
,
,…,
是
,
,…,
的任一排列,我们称
为这两组正实数的乱序和,
为这两组正实数的反序和,
为这两组正实数的顺序和.根据排序原理有
,即反序和≤乱序和≤顺序和.则下列说法正确的是( )
![](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/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdd14381552c8f3a896675effe1f4f4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b715e7842b95f654f16056a7c7f2abe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13423c094861baf4b759b7f3d8c3c226.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/473bae0bc5acfd6a486c1433c58fb369.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7936359df4c926b72b48c6fdae55f12d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b76f79be89b8c6227b68eded6b675546.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59e7c7a84a4bdb959e95536d0404ceb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b715e7842b95f654f16056a7c7f2abe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13423c094861baf4b759b7f3d8c3c226.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a71581a9193c69e5d4f644e6100ffd34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6214883c5996a384348d3052f05dd2b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6afad2abfe7cfccd0d634c1d8951121b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a357cbf7931e33754ae34bca26b98ac3.png)
A.数组![]() ![]() |
B.若![]() ![]() ![]() ![]() ![]() |
C.设正实数![]() ![]() ![]() ![]() ![]() ![]() ![]() |
D.已知正实数![]() ![]() ![]() ![]() ![]() |
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5 . 到
分别为
,求
的最小值.
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2024高三·全国·专题练习
解题方法
6 . 材料1.类比是获取数学知识的重要思想之一,很多优美的数学结论就是利用类比思想获得的.例如:若,
,则
,当且仅当
时,取等号,我们称为二元均值不等式.类比二元均值不等式得到三元均值不等式:
,
,
,则
,当且仅当
时,取等号.我们经常用它们求相关代数式或几何问题的最值,某同学做下面几何问题就是用三元均值不等式圆满完成解答的.
题:将边长为的正方形硬纸片(如图1)的四个角裁去四个相同的小正方形后,折成如图2的无盖长方体小纸盒,求纸盒容积的最大值.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/20/a7381e95-b28b-4dcb-a499-0831c39f7821.png?resizew=252)
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解题方法
7 . 已知
的值域为
.
(1)求实数
的值;
(2)判断函数
在
上的单调性,并给出证明;
(3)若
,求证
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3be8c296dba4a6442f262437f6671c80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a2ec965488c7e1cea085463c7731285.png)
(1)求实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9a475fec8ded321e10a6697319fb975.png)
(2)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4f3966052d4a779b6247fdf12f97cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d188ec2580e273ce87e51653a2177ee.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bf039c46a25e331446c6ee1e9af3c82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efb85ae535f90b3c125d86b439ab2562.png)
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解题方法
8 . 已知函数
为锐角,设
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ef6aac49c48e1aa12ff1fad74700225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/103506ffcbab54b205ab74449ada4248.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
9 . 不等式
中
的取值范围是
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/934d37c81b2266c7b86bcc11afaf5f91.png)
___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64f511d6169cf710fde7988bc598da36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5374dfeacbb54f1d1e9f14bc29fe406.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/934d37c81b2266c7b86bcc11afaf5f91.png)
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10 . 定义:(i)
表示x的最小值;(ii)
表示不超过x的最大整数.设a,b,c为正数,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3de36f09136a15d9f9ed3b0793d79860.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c179da37f00a4188877d6a803fae508.png)
A.0 | B.2 | C.3 | D.4 |
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