1 . 已知函数
,其中
,
.若点
在函数
的图像上,且经过点
的切线与函数
图像的另一个交点为点
,则称点
为点
的一个“上位点”,现有函数
图像上的点列
,
,…,
,…,使得对任意正整数
,点
都是点
的一个“上位点”.
(1)若
,请判断原点
是否存在“上位点”,并说明理由;
(2)若点
的坐标为
,请分别求出点
、
的坐标;
(3)若
的坐标为
,记点
到直线
的距离为
.问是否存在实数
和正整数
,使得无穷数列
、
、…、
…严格减?若存在,求出实数
的所有可能值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d634c247833c730fe1479bfd17fb854.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aff8d9b6533ff319420cdc5e8740b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b104090ea2ac34be58a76a4e0e95cb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df1d9b712b639c8b6809c9f3ae03706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34ac910421d6eab7694afa1e8e1027c5.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a882037b9ce104ecc496e0f31a139361.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b104090ea2ac34be58a76a4e0e95cb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f51e4cbda9f71ef70602e219bd93600.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df1d9b712b639c8b6809c9f3ae03706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32efe4eff75508cb93e828c735dcb695.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b104090ea2ac34be58a76a4e0e95cb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88c3a2f5b0702ea9fbb9dc8904579737.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff4d12362d4b8dd25813953e1c5a94b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82e260b088f071983f254ce8f5163fcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ae87824fc1b8133e45b96dd3d9f9ad5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e45801215e2e42c256860ce84a4dabf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01d065999821c48afebabb638a5a9cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
2 . 已知①设函数
的值域是
,对于
中的每个
,若函数
在每一处
都等于它对应的
,这样的函数
叫做函数
的反函数,记作
,我们习惯记自变量为
,因此
可改成
即为原函数的反函数.易知
与
互为反函数,且
.如
的反函数是
可改写成
即为
的反函数,
与
互为反函数.②
是定义在
且取值于
的一个函数,定义![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db6c5d65e0ccd27748fcbc420c6a2e22.png)
,则称
是函数
在
上的
次迭代.例如
,则
.对于一些相对复杂的函数,为求出其
次迭代函数,我们引入如下一种关系:对于给定的函数
和
,若函数
的反函数
存在,且有
,称
与
关于
相似,记作
,其中
称为桥函数,桥函数满足以下性质:
(i)若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e0e8a92152787274ed6e06a21ef1661.png)
(ii)若
为
的一个不动点,即
,则
为
的一个不动点.
(1)若函数
,求
(写出结果即可)
(2)证明:若
,则
.
(3)若函数
,求
(桥函数可选取
),若
,试选取恰当桥函数,计算
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e66c24e657d998beb013ad1fb311d33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f93e3581e920716e710e22b31006bdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f93e3581e920716e710e22b31006bdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63ced31d098cfb0cf14d906e97e6353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e66c24e657d998beb013ad1fb311d33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/955f3c2b80eebc3f88c804112e5f41f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/955f3c2b80eebc3f88c804112e5f41f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb87ce86e2c9a1a8188b03b74438fdd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb87ce86e2c9a1a8188b03b74438fdd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e66c24e657d998beb013ad1fb311d33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25ef38135b0e7906687d8a4918a4cb67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b161347f6a2fcfd9bf0acf1e8a03fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43737e3ca063dfc210d0c72924a4930.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/808bd2fc4b344e7669fca65b4fa122df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b161347f6a2fcfd9bf0acf1e8a03fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/808bd2fc4b344e7669fca65b4fa122df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b161347f6a2fcfd9bf0acf1e8a03fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db6c5d65e0ccd27748fcbc420c6a2e22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c110a1293773729278a214c7fe8d544e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a33cfe27fd2276a7c542f062c17b4d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/052ddf3664af9ab2990f3ea622997e00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/910ca4e4f009554b599eab90e1d94c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71567deb76e48f8a2424b06536cbe465.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a66b033ee7a03c7b3508583481465275.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1249f186df944244da02e1b8c754005.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
(i)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1249f186df944244da02e1b8c754005.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e0e8a92152787274ed6e06a21ef1661.png)
(ii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66f66a2b3d90f0d935d6c8ebaf675349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4f199ad3fad8657afa38f370b319a75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192bb45bd15b200f40b34377bc58905b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a33cfe27fd2276a7c542f062c17b4d85.png)
(2)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1249f186df944244da02e1b8c754005.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99315f5b2ae9bea18e06401b41d3780c.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/becd598a11b876d858728161a7a09705.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a33cfe27fd2276a7c542f062c17b4d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04ab7717944da2b6cc305b6a65f91408.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c85f9d9be0ba965ff7beb0e011267f29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bee7f1ccd52c7d526b6d466b970e769.png)
您最近一年使用:0次
解题方法
3 . 微分中值定理是微积分学中的重要定理,它是研究区间上函数值变化规律的有效工具,其中拉格朗日中值定理是核心,它的内容如下:
如果函数
在闭区间
上连续,在开区间
可导,导数为
,那么在开区间
内至少存在一点
,使得
,其中
叫做
在
上的“拉格朗日中值点”.已知函数
.
(1)若
,求函数
在
上的“拉格朗日中值点”
;
(2)若
,求证:函数
在区间
图象上任意两点
,
连线的斜率不大于
;
(3)若
,且
,求证:
.
如果函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30277e0be448b4955903e81e8795e45d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a4b04824a308519a61318a82aa97a05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30277e0be448b4955903e81e8795e45d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f3fcc5073759c73c7a63c8818eca5c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d11582cafaca7560189cf57e70f6a46d.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58423a31ef72d8d161b775090c9ed2c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bcc88c47bbbace2c56adced4f781b6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d0d09c61ed9d7289d948d2ab559657f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870ebc2f7aabb028024894568d749934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25a9a2cda21bc7d84330702878e61c63.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1dffe15ea2b4735ad2a274144301328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1310a7a80d1f8751a3f8cafe7f8c8b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098345d96f648ba45ad923a79e815496.png)
您最近一年使用:0次
名校
解题方法
4 . 给定自然数
且
,设
均为正数,
(
为常数),
.如果函数
在区间
上恒有
,则称函数
为凸函数.凸函数
具有性质:
.
(1)判断
,
是否为凸函数,并证明;
(2)设
,证明:
;
(3)求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fe1c31a81f198c443e71b83ca662939.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0376209b36fa0577a93f281dd68b86f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ced4181800832cf83f9dbe8dbeebada3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df0dd6144e9a30d1a063b690033c3f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee9fc6a26f68ea2ec181e18532659ddd.png)
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/301a7643aa976ee5b277abfd6b0c26a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047056c99b39c70fa40d3c8178e5b631.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0aec6fb84e2f7401f56146293b2e6289.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae3bd8d8090570b4f9cf779cea76570a.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/109abcd5418ef7b5757814817db1c973.png)
您最近一年使用:0次
解题方法
5 . 1799年,哥廷根大学的高斯在其博士论文中证明了如下定理:任何复系数一元
次多项式方程在复数域上至少有一根(
).此定理被称为代数基本定理,在代数乃至整个数学中起着基础作用.由此定理还可以推出以下重要结论:
次复系数多项式方程在复数域内有且只有
个根(重根按重数计算).对于
次复系数多项式
,其中
,
,
,若方程
有
个复根
,则有如下的高阶韦达定理:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68be203b2490ecce4c0e2eadeb5d911b.png)
(1)在复数域内解方程
;
(2)若三次方程
的三个根分别是
,
,
(
为虚数单位),求
,
,
的值;
(3)在
的多项式
中,已知
,
,
,
为非零实数,且方程
的根恰好全是正实数,求出该方程的所有根(用含
的式子表示).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e167b43045b3297248e334c41c621b8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b024d78f428194127b5534f948810def.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7230de53663c75658c58bbf206a0085.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bed25da42194b5a81d123933d5704f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd3759b3561834cdc5b499b91f3850d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b92b70365c63607daecdc8deb73ecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c83590c4a7ea5636843dd4b60c67cb40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68be203b2490ecce4c0e2eadeb5d911b.png)
(1)在复数域内解方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4800c5aa0e5b70b2141541cbd3853e34.png)
(2)若三次方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac603c0b3d1d7fd42bd50222b6ab94d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6755cd39b121a0dd2a14da8d43c1fff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ddb97874a62bb5530514a467d64af13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8079c5a2d8674d322f7abe6d4ef4a3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.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)
(3)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b024d78f428194127b5534f948810def.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cb3db0a99f86232e0cf3e55c789ea99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e2e2674707c28eddd3f3ab60f73f54f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c37d6353f394a5704a92113908a5c3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b92b70365c63607daecdc8deb73ecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
您最近一年使用:0次
名校
解题方法
6 . 柯西中值定理是数学的基本定理之一,在高等数学中有着广泛的应用.定理内容为:设函数
,
满足①图象在
上是一条连续不断的曲线;②在
内可导;③对
,
.则
,使得
.特别的,取
,则有:
,使得
,此情形称之为拉格朗日中值定理.
(1)设函数
满足
,其导函数
在
上单调递增,判断函数
在
的单调性并证明;
(2)若
且
,不等式
恒成立,求实数
的取值范围;
(3)若
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4562f3225c98cf5cb11b47d98c9cc9c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f9b92a1988f20c45e8ba3887eeb6b7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce5b83b652a50ea15c83c826d8fb52f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c38593523a246f9e59688f64444e0dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2a1212aca40e8dfbb97ae428c5d40a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4306fb6d5419322b4b7b9140e06e43a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c38593523a246f9e59688f64444e0dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/584ef8a5b63c5a2a80372865ac0cc0a0.png)
(1)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01bea8bf593f594c51fc7cc547482bee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acaa791feb147bd1a8bf5eb4f81a0cbc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a4b4a9b7f0a8c3de045fe903204800.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/432d77fe5ad3032d59a237dd94c8a638.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72e71b49ac6c97943138bed91aab6215.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d64f25e0020c3db48bb6a767afa98a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b19cf16fd398ad9782cd4f5149d0c76f.png)
您最近一年使用:0次
2024·全国·模拟预测
解题方法
7 . 函数与函数之间存在位置关系.已知函数
与
的图象在它们的公共定义域
内有且仅有一个交点
,对于
且
,
且
,若都有
,则称
与
关于点
互穿;若都有
,则称
与
关于点
互回.已知函数
与
的定义域均为
,导函数分别为
与
,
与
的图象在
上有且仅有一个交点
,
与
的图象在
上有且仅有一个交点
.
(1)若
,
,试判断函数
与
的位置关系.
(2)若
与
关于点
互回,证明:
与
关于点
互穿且
在
上恒成立.
(3)研究表明:若
与
关于点
互穿,则
与
关于点
互回且
在
上恒成立.根据以上信息,证明:
(
为奇数).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43db00e106c7d08a76a7ba71ca5e63d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d1e02162319786352c21176ce1e2dfb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68b7fda3b73e233090b9708b096ade24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c28e384ba050b238e11f7c74d3002aab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8549cf691ec1c21b1bf25152ca631ac5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b0bf4dc50ce06347d644da4910fc0d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43db00e106c7d08a76a7ba71ca5e63d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632fad7bc6013b0b9a7690caa90cb1d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43db00e106c7d08a76a7ba71ca5e63d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22add663bd26e87d972a10dc5fd9ada1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72265aa6a453a941dca4c9592b1e9c96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22add663bd26e87d972a10dc5fd9ada1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e609309bb5ec3d8524a75dcbdabd3f30.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ff6838d84b68c6f0d3b93b196d9b08d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab042ebfceebd0a1aa6b6bfe8180f8d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22add663bd26e87d972a10dc5fd9ada1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e609309bb5ec3d8524a75dcbdabd3f30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72265aa6a453a941dca4c9592b1e9c96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/835bd1fc94fae38fcc8a5f6d0be42cc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/574bdac28b1dcbf03f0fb903e8d0b49a.png)
(3)研究表明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22add663bd26e87d972a10dc5fd9ada1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e609309bb5ec3d8524a75dcbdabd3f30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72265aa6a453a941dca4c9592b1e9c96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/835bd1fc94fae38fcc8a5f6d0be42cc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/574bdac28b1dcbf03f0fb903e8d0b49a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cecf580720e09f57c9b58ab854e6ab7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
您最近一年使用:0次
名校
8 . 已知
与
都是定义在
上的函数,函数
图像上任意两点
,记
表示此两点连线的斜率.当
时,都有
,则称
是
的一个“T函数”.
(1)判断
是否为函数
的一个
函数,并说明理由;
(2)设
的导数为
,求证:关于
的方程
在区间
上有实数解;
(3)函数
的导函数存在记为
,即
导函数存在记为
,当
都有
,函数
是否存在T函数?若存在,请求出
的所有
函数;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e3a1467ecf286e3cadaf5aa006606f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c1dc007e36c78ab98df4cd2383b4c5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26d8dafc71b106f39f4e15442220897b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3861f863fc3d8703abf9e5bf97ef6117.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95a43b43650ed3473888a95607908644.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5ff3dc91272a2244b4d76056967e9a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/591c5b712dc14517e369be2345526fc7.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ff6838d84b68c6f0d3b93b196d9b08d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4c513ef355a637fff90a3371dc5328a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa852120429d7db38eb6266cf9d0d152.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4562f3225c98cf5cb11b47d98c9cc9c3.png)
(3)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d98da166934830f1cfdbcd48dbfea6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2287352f6b9c1ef9e35d2ac6670fcbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95e82f423c1c5d304766d1a22d72f042.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66692ec49a458f9e48c7315d03dfc37b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34367bca1e9f02459dc301e4881edbe5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/591c5b712dc14517e369be2345526fc7.png)
您最近一年使用:0次
9 . 已知无穷数列
是首项为1,各项均为正整数的递增数列,集合
.若对于集合A中的元素k,数列
中存在不相同的项
,使得
,则称数列
具有性质
,记集合
数列
具有性质
.
(1)若数列
的通项公式为
写出集合A与集合B;
(2)若集合A与集合B都是非空集合,且集合A中的最小元素为t,集合B中的最小元素为s,当
时,证明:
;
(3)若
满足
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86d70b1ef068e07c0ed707c17c11ffd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82079a5446d448fb1bea730b968d7e02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9652602f1d23494c53743efe03db6bdf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e577f08c801db946d97a024545bb5ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0380d25c8bccf9b2abdb668fb1bc5400.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/259af6f2d42a977dc6db0da888f6428a.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dec7ba4ecbcc20bfb5b7b3f473050eb0.png)
(2)若集合A与集合B都是非空集合,且集合A中的最小元素为t,集合B中的最小元素为s,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/682cbe4cd0d5cf5beb79d3ab89a117f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/997ba7c3da0821973b7f44d2ca07fcd1.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0d5a9b7c16226569966db27c11982f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46a70d32c64918aa4d1d9d3ce0bdbf7b.png)
您最近一年使用:0次
2024·全国·模拟预测
10 . 在参数估计的各种方法中极大似然估计法是应用最为广泛的一种估计方式,它广泛运用在金融、工程、生物制药等领域.把使样本事件发生概率最大的参数值,作为总体参数的估计值,就是极大似然估计.求极大似然估计的一般步骤:(1)由总体分布导出样本的联合概率函数(或联合密度);(2)把样本联合概率函数(或联合密度)中自变量看成已知常数,而把参数
看作自变量,得到似然函数
;(3)求似然函数
的最大值点(常转化为求对数似然函数
的最大值点);(4)在最大值点的表达式中,用样本值代入就得参数的极大似然估计值.已知服从正态分布
的样本中参数
的似然函数为
;服从二项分布的似然函数为
(其中
表示成功的概率,
为样本总数,
为成功次数),则下列说法正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c6eb532cf48aa754a0a0731a88c731b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c6eb532cf48aa754a0a0731a88c731b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c6eb532cf48aa754a0a0731a88c731b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29bcc248a7770a16fa10fc4602d71e0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97dc0c4d9a68ca8ce797958f9084fd4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a145eb64d3aff50cb9ced87c4703b7b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a7881f1f724495a233bdcebdd96ddf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
A.![]() ![]() |
B.参数![]() ![]() |
C.参数![]() ![]() |
D.二项分布中成功的次数与不成功的次数之比的极大似然估计值为![]() |
您最近一年使用:0次