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1 . 根据多元微分求条件极值理论,要求二元函数
在约束条件
的可能极值点,首先构造出一个拉格朗日辅助函数
,其中
为拉格朗日系数.分别对
中的
部分求导,并使之为0,得到三个方程组,如下:
,解此方程组,得出解
,就是二元函数
在约束条件
的可能极值点.
的值代入到
中即为极值.
补充说明:【例】求函数
关于变量
的导数.即:将变量
当做常数,即:
,下标加上
,代表对自变量x进行求导.即拉格朗日乘数法方程组之中的
表示分别对
进行求导.
(1)求函数
关于变量
的导数并求当
处的导数值.
(2)利用拉格朗日乘数法求:设实数
满足
,求
的最大值.
(3)①若
为实数,且
,证明:
.
②设
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4a1d0dba29a77dd111efcde543d6c1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc4c14935585e8fa61d032730867d771.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67b6f154c6b2de5695eb1807b98c2c63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/809615d1f91508e2c6c0cda7e592c479.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/244021f826099b18e31af1143597bba2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb5be11a5e6aaf00b2833930b198b4cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0203b006524305c3d8ee0b6c34cd872b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4a1d0dba29a77dd111efcde543d6c1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc4c14935585e8fa61d032730867d771.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1c3c1ed4fb65ab9505ad8078d8d0fb5.png)
补充说明:【例】求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d7ca0caa9933b7afd4bed2683140a07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aebdee8d81b048b5aa520f7e8ba56ff2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1e15a54c6122c695239107dd0901bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/244021f826099b18e31af1143597bba2.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b3d9ab2fcf15b94f33cb64f84ed906c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
(2)利用拉格朗日乘数法求:设实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c45d8122b61de13875003d00c002c5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de725a9fc66f67abbe0015131846a648.png)
(3)①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a14c388e1e2e5a2ff1ccf6caffbee0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd24c686fbaaa68705d654b880481ffe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e778f95c72fec00bfbbc63e6dfd0c460.png)
②设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/497d269c30eec393e3f0e877ddbe2983.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ade042c085bbad8aeaf111b9f4c33408.png)
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2 . 已知
.
(Ⅰ)若
时,
的解集为
,解不等式
;
(Ⅱ)若
,
,解关于
的不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4e73001005c51065ad1315be7a4175d.png)
(Ⅰ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e520c1ab44faaa476a5f3f6181db0f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c441f43aa65684e2d5e04cd081461cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54bc5feff8fc3d1145463d520a7f650c.png)
(Ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d464f6926d5666db5836843f3d73dd71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e768f6d07a30c490a1011a8256548bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6000b174147cec2de26041837aec1b3.png)
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2020-07-27更新
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3卷引用:江苏省盐城市东台市创新学校2020-2021学年高二上学期9月检测数学试题
3 . 下列说法正确的是( )
A.![]() |
B.![]() |
C.若![]() ![]() |
D.方程组![]() ![]() |
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4 . (1)计算:
;
(2)先化简,后求值:
,其中
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2393fbd54f9528459f5f5cbe0290c1a.png)
(2)先化简,后求值:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8998c5f571a64b637a2366ecb19a37e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65397f11ea8af736f38debadf420c4a.png)
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5 . 已知点
和直线
,则点
到直线
的距离证明可用公式
计算.
例如:求点
到直线
的距离.
解:
直线
,其中
,
.
点
到直线
的距离为:
.
根据以上材料,解答下列问题:
(1)求点
到直线
的距离;
(2)已知⊙
的圆心
坐标为
,半径
为
,判断⊙
与直线
的位置关系,并说明理由:
(3)已知直线
与
平行,求这两条直线之间的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7775aa57ca0e62216f3039ed88dceed0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c15fb18163df0690365a0d2e7ee88f5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c15fb18163df0690365a0d2e7ee88f5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0682ce7c7d01d65347c659227e6c3e15.png)
例如:求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a42451bdbef6c82dbaf8e06f0614794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/960c22d0509ff3a0d4620afe187b196a.png)
解:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/960c22d0509ff3a0d4620afe187b196a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367e788c32187ae2cc97aaa24da1d40d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ab1c19b66cda3fb899f06d9a25e973c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a42451bdbef6c82dbaf8e06f0614794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/960c22d0509ff3a0d4620afe187b196a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46ed48c24e5697d14fe19abf3586fa6f.png)
根据以上材料,解答下列问题:
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c2f21b1baf0624482fd41d7ba390341.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e235d7dd12f948f5ffb2e5afddc95612.png)
(2)已知⊙
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb0e705301752424a492f6277ed7774e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5627cec233ab4cd6ea8a864e220a6946.png)
(3)已知直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1d7df623642896d720d6956ed1f0ff6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a515853c22f0145b36c512079134dd5.png)
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6 . 对于三次函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19951f3364fb04433feed743bc37975d.png)
,给出定义:设
是函数
的导数,
是
的导数,若方程
有实数解
,则称点
为函数
的“拐点”,同学经过探究发现:任何一个三次函数都有“拐点”;任何一个三次函数都有对称中心,且拐点就是对称中心,若
,请你根据这一发现,求:(1)函数
的对称中心为___________ ;(2)计算![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cb8378ae701d9f5a5d66da3cf62c65d.png)
___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19951f3364fb04433feed743bc37975d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f5bb89c3ad435f1ef59307b174105ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10acd6d864583617dd3e71240bf0c857.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df1fa6ca9eb7cea9131dad36db6a0ac6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43db00e106c7d08a76a7ba71ca5e63d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7ed163db8f836e5399406e6d8a7fbd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cb8378ae701d9f5a5d66da3cf62c65d.png)
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10卷引用:江苏省盐城市大丰区新丰中学2022-2023学年高二上学期期末数学试题
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7 . 已知二次函数
,当
时,
;当
,
.
(1)求
,
的值;
(2)解关于
的不等式:
且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78222043de986bca085f490521326e09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f849096465001a54b6d2243c95263d07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebadf71a3c73c1d82ae821018a7f67c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e627dd89b76b55ea59f5fc9bec5ea13f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebfdecc7f8089cb23c20d0a93ee1b601.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
(2)解关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bac106969855ae6ba2136b6ebb66e545.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfc595f8b5666b09cbf06eb8389064cf.png)
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解题方法
8 . 函数
是定义在
上的奇函数,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6c10761f7a564f78488bde9108dd81.png)
(1)求函数
的解析式;
(2)判断
在
上的单调性,并用定义证明;
(3)解关于
的不等式
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eaa9cabdc771e2d2476a537b6e0d126b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ba6e6f0b3a0650b0a85aa419c5347d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6c10761f7a564f78488bde9108dd81.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ba6e6f0b3a0650b0a85aa419c5347d4.png)
(3)解关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/623575dd80affb5dca621e9a76f51eca.png)
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9 . 已知函数
,
.
(1)求函数
的最小正周期和单调递减区间:
(2)设函数
,解关于
的不等式
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2166a0b45aa4b7449ca55c37fb9b9697.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb63478132d4c1fef3c17e591919da83.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12c013a1f74918e84d1af6ee4a7408d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1be81ca31873c4ce92e9eccc8bfde1bb.png)
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名校
10 . 关于
的不等式
.
(1)若不等式的解集为
或
,求
的值;
(2)解关于
的不等式
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c73ccb709b15f8263041a1989a505da.png)
(1)若不等式的解集为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1eabfba9a63ab75e3935f1c0e9b8e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c751a6e261e30b9db5a369bc7ea36d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)解关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c73ccb709b15f8263041a1989a505da.png)
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2023-01-05更新
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江苏省东台市创新学校2017-2018学年高二9月月考数学试题2014-2015学年湖北省宜昌市金东方高级中学高一6月月考数学试卷2014-2015学年山西省大同一中高一下学期期末数学试卷辽宁省抚顺市第十九中学2016-2017学年高二上学期期中考试数学试题【全国百强校】重庆市綦江中学2017-2018学年高一下学期第三学月考试数学试题江苏省扬州市高邮中学2020-2021学年高二上学期9月月考数学试题山东省菏泽市郓城第一中学2021-2022学年高三上学期第一次阶段性检测数学试题天津市红桥区2016-2017学年高一下学期期末数学试题广西钟山县钟山中学2021-2022学年高一上学期第一次月考数学试题(已下线)河南省信阳市信阳高级中学2023-2024学年高一下学期3月月考(一)数学试题