1 . 设
,用
表示不超过x的最大整数,则
称为取整函数,取整函数是德国数学家高斯最先使用,也称高斯函数.该函数具有以下性质:
①
的定义域为R,值域为Z;
②任意实数都能表示成整数部分和纯小数部分之和,即
,其中
为x的整数部分,
为x的小数部分;
③
;
④若整数a,b满足
,则
.
(1)解方程
;
(2)已知实数r满足
,求
的值;
(3)证明:对于任意的大于等于3的正整数n,均有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb63478132d4c1fef3c17e591919da83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f161c2a3717f1b6c62d0d7dae0b606.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5b7f26fe1977bda9de200debe99f020.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5b7f26fe1977bda9de200debe99f020.png)
②任意实数都能表示成整数部分和纯小数部分之和,即
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9643772929ed7ee674ae68adb5381265.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f161c2a3717f1b6c62d0d7dae0b606.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/216921512381b9ebbb9cc59ecc9eb427.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5f7e2f76a9643572acc81394e9b965a.png)
④若整数a,b满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4513fc3f11c7030d7c83294335de57f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1e38f0d07ed41a7e373b3f8a281eef.png)
(1)解方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ab4334cd34a187b787278e1b2cb214b.png)
(2)已知实数r满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ed5c396204fbca3ef755668b277f6a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/904778775ee8cf551428f21b5b0ca915.png)
(3)证明:对于任意的大于等于3的正整数n,均有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39219116316e31189df7d04d6b9f428b.png)
您最近一年使用:0次
名校
解题方法
2 . 已知函数
.
(1)求不等式
的解集;
(2)若
且满足
,记
是
的最大值,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22c2e033105f17e4ea375d28464413ab.png)
(1)求不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e99f6241f03f76761403af0c53d3a0f1.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c457d9c7bbd4fb8d54c032565a2667b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1fa1634d00a91a067feb12dcf03d633.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/073aaf052fc8858629986a9dad40ff88.png)
您最近一年使用:0次
2023-04-04更新
|
391次组卷
|
3卷引用:广西梧州市苍梧中学2023届高三5月份高考数学模拟试题
11-12高一上·北京·期中
名校
解题方法
3 . 设函数
的定义域是
,且对任意正实数x,y都有
恒成立,已知
,且当
时,
.
(1)求
的值;
(2)判断
在区间
内的单调性,并给出证明;
(3)解不等式
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25bea6d14c16f7c06e4e028f36131360.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af9da4fdfdddc259dcef9fdd4b826b64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fde64f4d3c38e43fbdee24eadc4b0dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c73a98c1b3504e09bfbe0db849b0d24.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63fef5f357f94e1e162cc47a99f9ab1e.png)
(2)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
(3)解不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cc574d99c154e7acf0e512c4c727d84.png)
您最近一年使用:0次
2022-11-22更新
|
1079次组卷
|
14卷引用:广西北流市2020-2021学年高一高中“农信杯”教学质量调研检测数学试题
广西北流市2020-2021学年高一高中“农信杯”教学质量调研检测数学试题(已下线)2011年北京市101中学高一上学期期中考试数学(已下线)第二章 3 函数的单调性(一)(课时作业)-2018版步步高学案导学与随堂笔记数学(北师大版必修1)海南省东方市民族中学2019-2020学年高一上学期期中数学试题安徽省滁州市民办高中2019-2020学年高一上学期期末数学试题(已下线)[新教材精创] 5.3 函数的单调性练习-苏教版高中数学必修第一册江苏省南通市海门市包场高级中学2020-2021学年高一上学期10月学情调研数学试题(已下线)卷09 函数的概念与性质 章末复习单元检测(难)-2021-2022学年高一数学单元卷模拟(易中难)(2019人教A版必修第一册)(已下线)专题3.1 抽象函数初步 A卷-2021-2022学年高一数学单元卷模拟(易中难)(2019人教A版必修第一册)辽宁省大连市第八中学2021-2022学年高一上学期期中数学试题河南省实验中学2022-2023学年高一上学期期中数学试题辽宁省辽东区域共同体2022-2023学年高一上学期期中联考数学试题湖北省武汉市黄陂一中盘龙校区2022-2023学年高一上学期11月适应性考试数学试题湖南省邵阳市第二中学2023-2024学年高一上学期期中数学试题
4 . 已知函数
,
.
![](https://img.xkw.com/dksih/QBM/2022/2/21/2921392171024384/2926939060887552/STEM/b043e32e-ccc6-4436-b29d-f6d4ecb99a85.png?resizew=235)
(1)画出
的图象,若
与
的图象有三个交点,求实数
的取值范围;
(2)已知函数
的最大值为
,正实数
,
,
满足
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e99fc9c920cad0fd4fab2b56fb18a300.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
![](https://img.xkw.com/dksih/QBM/2022/2/21/2921392171024384/2926939060887552/STEM/b043e32e-ccc6-4436-b29d-f6d4ecb99a85.png?resizew=235)
(1)画出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/408a9272589ff1809c137869ce8673a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](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/f20421354eeae0a1b34ba40a57992402.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbadbad21e77939800df4772b977103e.png)
您最近一年使用:0次
2022-03-01更新
|
801次组卷
|
5卷引用:高考广西桂林、崇左市2022届高三5月联合模拟考试数学(文)试题
高考广西桂林、崇左市2022届高三5月联合模拟考试数学(文)试题广西桂林、河池、来宾、北海、崇左市2022届高三5月高考联合模拟考试数学(理)试题贵州省铜仁市2022届高三适应性考试数学(理)试题(—)贵州省贵阳市2022届高三适应性考试(一)数学(理)试题(已下线)重难点07 选考极坐标与参数方程、不等式 -2022年高考数学【热点·重点·难点】专练(全国通用)
名校
5 . 已知函数
.
(1)求
的值域;
(2)若
的最大值为m,正实数工x,y,z满足
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45c2d8dbbbd66fb7696d633a65876849.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f6bfdb24ecf5da863405c2b40936ff9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f6bfdb24ecf5da863405c2b40936ff9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3576df052a24144ab6b0f7a097a1261.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e83f21a6bf2c84fc44a1dad44799a481.png)
您最近一年使用:0次
2022-01-16更新
|
736次组卷
|
5卷引用:广西柳州市2022届高三第二次模拟考试数学(文)试题
名校
解题方法
6 . 已知函数
.
(
)判断并证明函数
在
的单调性.
(
)若
时函数
的最大值与最小值的差为
,求m的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cafe688199de30368f9c5e2272567511.png)
(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c1c3e201160759b1e93d70ad439d33d.png)
(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2425497f56c3bdc9a8d7cde18e41d11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
您最近一年使用:0次
2020-11-23更新
|
396次组卷
|
4卷引用:广西玉林市北流高中、陆川中学、岑溪中学、容县高中四校2020-2021学年高一年级12月联考数学试题
解题方法
7 . 已知
是奇函数.当
时,
.
(1)当
时,求
的解析式;
(2)用定义证明:
在
上是减函数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b68dda9b3f4d479548dcc39c07ac5f52.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e541ea2f855f981c96207070683d388.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)用定义证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8938db94f49dcbe0c383fba0241bb0da.png)
您最近一年使用:0次
解题方法
8 . 已知函数
,
.
(1)当
时,求不等式
的解集;
(2)若
,求函数
的单调递增区间;
(3)求证:当
时,对于任意两个不等的实数
,均有
成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64263fe2ca48e694c87496d61e63fb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13ea47a1bfc7b4e1f8383af10355b2ad.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9795048c0752ac17568f4f1d98d18299.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20849c00c47cbdc43f18d53341b6c4e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71607511fdd4faa9e832345ceb2a817d.png)
(3)求证:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50c4e6c93e32316aea38bfab92b3344f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bcd66b9267dd8cc2e661e3d0d411acf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b24ff3b1b33d0780307e5d6e003d6ad.png)
您最近一年使用:0次
9-10高三·湖南湘潭·阶段练习
名校
解题方法
9 . 已知二次函数
对
都满足
且
,设函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/079f277788b447aabb0b527020dc20a4.png)
(
,
).
(1)求
的表达式;
(2)若
,使
成立,求实数
的取值范围;
(3)设
,
,求证:对于
,恒有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ad5fe274cfc8da2dacfb65801f344ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb38ca84b4eadbe4eaa09bb5c778d912.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1feffb0bb2658090edd0b2f9f2721fc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/079f277788b447aabb0b527020dc20a4.png)
(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40e72f6b2ef3329828cb8fc873eeba7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5bf7d03f075e1c0a67d02a56ddd6611.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dc9ede2e55724383dd1093fc7fcdb59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86aada6c0797463dd75408a0ad45c43e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e07a024c9ae1e811ea066430c02fd62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfd1b5a2a71ae12061e768a1814f536a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/950e0e6b300b63f5424deaa89734811f.png)
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