1 . 定义:若函数
与
的图象在
上有且仅有一个交点,则称函数
与
在
上单交,此交点被称为“单交点”.已知函数
,
,
.
(1)讨论函数
的单调性;
(2)当
时,
(i)求证:函数
与
在
上存在“单交点”
;
(ⅱ)对于(i)中的正数
,证明:
.
![](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/b3eb5935678e432e6f1f3180bfdb3175.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/b3eb5935678e432e6f1f3180bfdb3175.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30ba24231723af1ea3d94be78053998f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22dd8b3dc4c609bab82d356a5cc2208d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62e19cdacdd4a47291e4621a8c167efc.png)
(1)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89e455f4e6c97270bd28f207b89df5fa.png)
(i)求证:函数
![](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/d562dc22dfb3b81d0c3f88b54d063c2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43db00e106c7d08a76a7ba71ca5e63d1.png)
(ⅱ)对于(i)中的正数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33e33f6cdfee603b548e158bcb1f82df.png)
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2 . 已知函数
,其中
.
(1)若
,证明:
时,
;
(2)若函数
在其定义域内单调递增,求实数
的值;
(3)已知数列
的通项公式为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc918d83961931831f58ee6ee88ce37f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10ede78fd7ac619ea597856254bb5d75.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3647a896689efaec8ae89cad1cd845d5.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4633de9335d15d7685bdecb007a3678c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd30509fe23160914e2cea22efe4b101.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ccc81f3cbaab5987151e4235b3600f8.png)
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2024-06-08更新
|
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2卷引用:安徽省蚌埠市2024届高三第四次教学质量检查考试数学试题
2024高三·全国·专题练习
3 . 已知
,函数
有两个零点,记为
,
.
(1)证明:
.
(2)对于
,若存在
,使得
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655c46b33730f3a29b9ec3024df71375.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17de16980dc347680c23b17153ef1232.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2aba77b36579eeccb98cdc308ce92bc8.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc043d78e4c9ad2281754d6c1cac8791.png)
(2)对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eedf333393bdf56f8b428e9a7d2eb3de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7443588709e037203d0962bc5b3c705.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c403614594d401cf38ebc4d48c2f47f3.png)
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解题方法
4 . 已知函数
.
(1)证明:
;
(2)设函数
,若
恒成立,求
的最小值;
(3)若方程
有两个不相等的实根
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc36a3c21811a9754a537062a73f43e6.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e799e937076aa5a7dcd51cdc0f40f6b0.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5706e65074de43ba1d3b0f5861646e1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa18838a13fda4e45612c32cdf98b71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(3)若方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9c0d827ef8598ba6b70b34b2bdcd1e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f98def21c9ea5780553a3dfb46d455f.png)
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5 . 已知函数
.
(1)求函数
的单调区间;
(2)若函数
的两个极值点分别为
,证明:
;
(3)设
,求证:当
时,
有且仅有2个不同的零点.
(参考数据:
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d797a2db981447da3e604690da4afca.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dabb97b3033a9915c9016df81df91b94.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ade931035db1a9b2f6f96ab9133148b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd60e69dac32dc020aacf5df042e5f2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ec068ca23b1226af6b27e20d57b87e9.png)
(参考数据:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74517b26a5a02c30c94f031d52d63014.png)
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解题方法
6 . 若函数
在定义域内存在两个不同的数
,同时满足
,且
在点
处的切线斜率相同,则称
为“切合函数”
(1)证明:
为“切合函数”;
(2)若
为“切合函数”,并设满足条件的两个数为
.
(ⅰ)求证:
;
(ⅱ)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/859458471c86ae39e0cc42d2d960d03e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbcc25bee0bd3ceeb3e8d0573f34b6b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a87b4c3b6486ddc142457f3781d898d8.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a5ca0a482b48b476356bf5e2c502810.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
(ⅰ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3a0b39ed179340810fea23d244406ce.png)
(ⅱ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65885209eb867c87729188328ae03261.png)
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2024-05-12更新
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2卷引用:重庆市名校联盟2023-2024学年高三下学期第一次联考数学试题
7 . 已知椭圆
经过点
,且焦距为
.
(1)求椭圆
的方程;
(2)设椭圆
的左、右顶点分别为
、
,点
为椭圆
上异于
、
的动点,设
交直线
于点
,连接
交椭圆
于点
,直线
的斜率分别为
.
①求证:
为定值;
②证明:直线
经过
轴上的定点,并求出该定点的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38387ba1cadfd3dfc4dea4ca9f613cea.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23d29646155e27b172ecdf263e2d702.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b1ec05e3cec27677ded7b4aecaa62d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5671fb25040a712a49e8c8148d67d300.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/706325cc86b99fe9955185aa92a8fcab.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d53a52aebd885294e323ee90c9b5382.png)
②证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
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解题方法
8 . 设实系数一元二次方程
①,有两根
,
则方程可变形为
,展开得
②,
比较①②可以得到![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71e360b7ba27dfc3e5d401027d5bd8a5.png)
这表明,任何一个一元二次方程的根与系数的关系为:两个根的和等于一次项系数与二次项系数的比的相反数,两个根的积等于常数项与二次项系数的比.这就是我们熟知的一元二次方程的韦达定理.
事实上,与二次方程类似,一元三次方程也有韦达定理.
设方程
有三个根
,则有
③
(1)证明公式③,即一元三次方程的韦达定理;
(2)已知函数
恰有两个零点.
(i)求证:
的其中一个零点大于0,另一个零点大于
且小于0;
(ii)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c3f3db6b7c682450309a6ccba5ac5a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
则方程可变形为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/455d33fcfd9a59d6b374e9d25888cd2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e42b42152492cbdfec62c7a02be4055.png)
比较①②可以得到
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71e360b7ba27dfc3e5d401027d5bd8a5.png)
这表明,任何一个一元二次方程的根与系数的关系为:两个根的和等于一次项系数与二次项系数的比的相反数,两个根的积等于常数项与二次项系数的比.这就是我们熟知的一元二次方程的韦达定理.
事实上,与二次方程类似,一元三次方程也有韦达定理.
设方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ddc5e3c2c7c6f4d2d0ab396b65679a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05b8ec9d4206ea66a02de5c4a1e1e911.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb8d75a2417827b2c5b09ba9385fe252.png)
(1)证明公式③,即一元三次方程的韦达定理;
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2b5e4746c2bd0afb279630698afd3a0.png)
(i)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274a9dc37509f01c2606fb3086a46f4f.png)
(ii)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20d6fc9b90f370fbb27552876b650f8f.png)
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9 . 已知函数
,曲线
在点
处的切线为
,记
.
(1)当
时,求切线
的方程;
(2)在(1)的条件下,求函数
的零点并证明
;
(3)当
时,直接写出函数
的零点个数.(结论不要求证明)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2be6b7c590b12db1b6cbe451ad18c4ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd817a1014876a72ad1971548ed6f52c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43db00e106c7d08a76a7ba71ca5e63d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db57c256dac51842864d269d5cdab520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa662f0273f0921c1fa4727f632395.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/909736dad505d81be43aef91e6309bf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(2)在(1)的条件下,求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a93aef9c6c4b64df420c39ef19d1551.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efe8dc8e5def7d46b88535453ae1fd96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
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10 . 已知函数
.
(1)判断并证明
的零点个数
(2)记
在
上的零点为
,求证;
(i)
是一个递减数列
(ii)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58ae99833cb675e5e36c58f345eb03e7.png)
(1)判断并证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d64af919a56a107e0fc0a417e481648.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d64af919a56a107e0fc0a417e481648.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870ebc2f7aabb028024894568d749934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
(i)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1fd18a909cecbaee7115d6b15631d83.png)
(ii)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/167f4f8fd4d3de714b87f05e57a3ba3b.png)
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2024-06-04更新
|
557次组卷
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2卷引用:湖南省长沙市第一中学2024届高三下学期模拟考试数学试卷(一)