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解题方法
1 . 拉格朗日中值定理是微积分学的基本定理之一,它与导数和函数的零点有关,其表达如下:若函数
在区间
连续,在区间
上可导,则存在
,使得
,我们将
称为函数
在
上的“中值点”.已知函数
,
,
.
(1)求
在
上的中值点的个数;
(2)若对于区间
内任意两个不相等的实数
,
,都有
成立,求实数t的取值范围.
(3)当
且
时,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.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/ee49cd415b686374189f90102d23ef7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdcb1944408a1f36d0123e093ea2f0ad.png)
![](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/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ff6838d84b68c6f0d3b93b196d9b08d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdbefd68d090abfa266cc874d4121f24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4633de9335d15d7685bdecb007a3678c.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46be55c8f2760d6db125f46691a3de48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
(2)若对于区间
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e938b003ed30316afc6163e1f856c3c.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fff6e7e2b9f2b68b1647f6350b98dc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a369ce3949b2bd2747a48054f7b951c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffa5572b8342e2c014a0cbdce7bd2dae.png)
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解题方法
2 . 英国数学家泰勒发现的泰勒公式有如下特殊形式:当
在
处的
阶导数都存在时,
.注:
阶导数指对一个函数进行
次求导,
表示
的2阶导数,即为
的导数,
表示
的
阶导数,
为自然对数的底数,
,该公式也称麦克劳林公式.设
,根据以上信息,并结合高中所学的数学知识,解决如下问题:
(1)利用泰勒公式求
的近似值;(精确到小数点后两位)
(2)设
,证明:
;
(3)证明:
(
为奇数).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6368fec0c2c25db7c29b014d60270e97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fb7618da716747c7cf514bbd1c58ad2.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/10acd6d864583617dd3e71240bf0c857.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35993bd1db970330494665d925c0be7a.png)
![](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/a6696028290bbaddf628d64bad0ed95b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11204e2fb6e560bf7a4ca26eaebfc526.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c78478b44ff22e088fd8e6522c5d78a2.png)
(1)利用泰勒公式求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25b6bad45bbc4b4c4cde24e16512c098.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66692ec49a458f9e48c7315d03dfc37b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8586154d8c4fb5fef893d39a7701f921.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48181c310080dbcf23704a76023adbc4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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解题方法
3 . 若
,关于
的不等式
恒成立,则正实数
的最大值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36fac79eb549c72549f3718d0c80bc92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2024-05-30更新
|
1358次组卷
|
3卷引用:山东省菏泽市定陶区第一中学2023-2024学年高二下学期5月月考数学试题
山东省菏泽市定陶区第一中学2023-2024学年高二下学期5月月考数学试题广东省广州市华南师范大学附属中学2024届高三下学期高考适应性练习(4月)数学试题(已下线)第3题 妙解指对函数最值(压轴小题)
名校
4 . 已知函数
.
(1)当
时,求
的单调区间;
(2)当
时,
恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1996957bc581e5d03c0f733233e9956.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b269c11e6f9dbbf1a1efcda572f13ea8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1c8abdaf1a883ca6d2fed14f8341696.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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名校
解题方法
5 . 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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6 . 已知函数
,
.
(1)当
时,求
在
处的切线方程;
(2)求
的单调区间:
(3)若
,
,使得
,求a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81d4ffe83140243af103c4d806e3ed1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7ef7094c6c60a57532334ee3e8b4afe.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da60d3fead7b237c07e817a3801cafe2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d41f25aab93b2ceb283fca684e9b8d4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a8c626afbc95213849e8f122d9b1a13.png)
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7 . 已知函数
.
(1)当
时,若
有两个零点,求实数
的取值范围;
(2)当
时,若
有两个极值点
,求证:
;
(3)若
在定义域上单调递增,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f56de34e94fca24318fa9976625fadb5.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/143b917df0520097be222accbddf9394.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4d795709b0abcf47bceec2250f2f9b.png)
![](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/41af6fa656b452c5f92816790b499201.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51442bf0ef97e173588b54ee77db26b5.png)
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解题方法
8 . 已知函数
,
.
(1)求函数
在区间
上的最大值;
(2)若
,
恒成立,求实数a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfcea74d330997ee9c92a223c0335851.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10bbdef421c976962a270a2beabbad91.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6c1756b564bf1d998d8179637011c88.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6422b9c2e93a91fe9e39ce4d9dabb0fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1000d36b729d9895dab2e60ee7df50b8.png)
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9 . 已知函数
.
(1)讨论函数
的单调性;
(2)当
时,若满足
,求证:
;
(3)若函数
,当
时,
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e20a21999ea818acdfb48d3641f70d3b.png)
(1)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbdc729607cf42c430488ff4bd2cd4f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fecfe7cc8dc611725c443293a3c2f377.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/556eac935a69ae56fb1d63bee5a1e5dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85187c85826beeca12137805293fff77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2024-05-27更新
|
894次组卷
|
3卷引用:河北省保定市六校联盟2023-2024学年高二下学期4月期中联考数学试题
名校
10 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b455c09ae609a3a2a24fe0999547e4a.png)
,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b455c09ae609a3a2a24fe0999547e4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41d5b338ddbc0bac8c8f25e036314fe1.png)
A.当![]() ![]() |
B.若经过原点的直线与![]() ![]() ![]() |
C.若![]() ![]() ![]() ![]() |
D.若![]() ![]() ![]() ![]() |
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