1 . 已知定义在
上的函数
的表达式为
,其所有的零点按从小到大的顺序组成数列
(
).
(1)求函数
在区间
上的值域;
(2)求证:函数
在区间
(
)上有且仅有一个零点;
(3)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7d9eebb0705256305ab3bf28898fffc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9df075cd20f79486d88d80ee12fc897d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0b6a6e136f1f05417c93473d27a5efe.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71163f419555f2ed76075c8ff659fbfc.png)
(2)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85fa35905c7193c20799ed7b925b358a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0b6a6e136f1f05417c93473d27a5efe.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0710952d6f8d8f6c0de73c42f4301c79.png)
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名校
解题方法
2 . “拐点”又称“反曲点”,是曲线上弯曲方向发生改变的点.设
为函数
的导数,若
为
的极值点,则
为曲线
的拐点.
已知函数
有两个极值点
,且
为曲线C:
的拐点.
(1)求a的取值范围;
(2)证明:C在Q处的切线与其仅有一个公共点;
(3)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85be63b7da7f1174c96176de8d1ecc9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85be63b7da7f1174c96176de8d1ecc9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35ce95d0450bc59111b516c56586cb78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/661249bf6499017f9e5e03db3fcd93d0.png)
已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4d05faec455cea37e004e18cfb7e290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1c12d99bdf82674ac9a1edceff81d54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
(1)求a的取值范围;
(2)证明:C在Q处的切线与其仅有一个公共点;
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e804ae37438267dd3a4b9c26d3d7c33.png)
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名校
解题方法
3 . 已知椭圆
的离心率为
,直线
截椭圆
所得的弦长为
.
(1)求椭圆
的标准方程;
(2)设直线
与
轴交于点
为粗圆
上的两个动点、且均位于第一象限(不在直线
上),直线
、
分别交椭圆于
两点,直线
分别交直线
于
两点.
①设
,试用
表示
的坐标;
②求证:
为线段
的中点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0c118b51ab426bc1c1b56179094f146.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/707ea658f3a9359f5740d5aab48f7948.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95bacae35b6e16a0a33c2bdc6bc07df7.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
(2)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/707ea658f3a9359f5740d5aab48f7948.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbc2dfda09549eac62c6f0c47d70625a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/707ea658f3a9359f5740d5aab48f7948.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63a253c7fdf589ee3dece13d5b5b5732.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8f90eb172dbd2ff7ae6f705801c0737.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ba48366317ebea1c9dd5e4e67e03092.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/707ea658f3a9359f5740d5aab48f7948.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cf9b288c48c73463a2f214f02b6952a.png)
①设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c663466d641b5fdfef1e529d6c330ecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eaf2102b730fe50c8681f1a6fafe67af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/166afeb61d5a80366a8ae29c912cd644.png)
②求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
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2024高三·全国·专题练习
解题方法
4 . 若
有两个极值点
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2bf442d1f65901dbe5059a1bd78e346.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/546a1ee9369c1c238e3e9ff1bb4a236e.png)
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2024·全国·模拟预测
5 . 在平面直角坐标系xOy中,已知圆A:
,点
,点P为圆A上任意一点,线段BP的垂直平分线和半径AP所在直线相交于点Q,当点P在圆上运动时,点Q的轨迹为C.
(1)求C的方程.
(2)斜率存在且不为0的直线l与C交于M,N两点,点D在C上.从下面①②③中任选两个作为已知条件,证明另外一个成立.
①
轴;②直线l经过点
;③D,B,N三点共线.
注:若选择不同的组合分别解答,则按第一个解答计分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/550d183b05000722c74baf25eb4a6741.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2c77a42750684cb6157c2c7fb9422a3.png)
(1)求C的方程.
(2)斜率存在且不为0的直线l与C交于M,N两点,点D在C上.从下面①②③中任选两个作为已知条件,证明另外一个成立.
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4914b846a985658a528ab9d70ccc7c30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f28491f7ef64389d62b0e1574ab56429.png)
注:若选择不同的组合分别解答,则按第一个解答计分.
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解题方法
6 . 已知函数
,
.
(1)求证:函数
存在单调递减区间,并求出该函数单调递减区间
的长度
的取值范围;
(2)当
时,
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f43d3391304d235789fb72d6e21f2e6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72ac49ab7c8001c209b8611b9ea40d85.png)
(1)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4562f3225c98cf5cb11b47d98c9cc9c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13502d46b8563c54c09b29b20b3006a4.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2fb40a36a293471742ce75f6b9635b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92231ba0e780c53511c9c386b60ec6ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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7 . 点
是椭圆
的左右顶点,若过定点
且斜率不为0的直线与椭圆
交于M,N两点,求证:直线AM与直线
的交点在一条定直线上.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e0b4bbfa0ed04cd3c2454d99d64e29c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7a999c36de5c9a9ce876a4a56fa34c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7785afeeaf274892253d04b4f693b367.png)
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2024高三·全国·专题练习
解题方法
8 . 已知函数
且
.
(1)求
;
(2)证明:
存在唯一的极大值点
,且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f450df1f33890352e96f0dd0d859cc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22e38c541dec8fce1d26886e5ef7d21f.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bc94b241dbf16e8adcbc0524eb89cbf.png)
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2024·全国·模拟预测
解题方法
9 . 已知抛物线C:
的准线方程为
.
(1)求抛物线C的方程;
(2)若斜率为1的直线l交抛物线C于A,B两点,点P,Q在C上且关于直线l对称,求证:A,B,P,Q四点共圆.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05f3023b526438933a05bed6f671730d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f32ae4ac252986a77c07b980d375a3fd.png)
(1)求抛物线C的方程;
(2)若斜率为1的直线l交抛物线C于A,B两点,点P,Q在C上且关于直线l对称,求证:A,B,P,Q四点共圆.
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