名校
解题方法
1 . 在
中,
,
,
对应的边分别为
,
,
,
.
(1)求
;
(2)奥古斯丁·路易斯·柯西(
,
年
年),法国著名数学家柯西在数学领域有非常高的造诣.很多数学的定理和公式都以他的名字来命名,如柯西不等式、柯西积分公式.其中柯西不等式在解决不等式证明的有关问题中有着广泛的应用.现在,在(1)的条件下,若
,
是
内一点,过
作
,
,
垂线,垂足分别为
,
,
,借助于三维分式型柯西不等式:对任意
,
,
,有:
,当且仅当
时等号成立.求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3818a2c9919d358b4c3713396093822b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/febc9a89d0d1c97b88c0f4acd32b4e67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/194741f4d2ae7ee44cafca780361446a.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)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00378760a6e82ce9cedf439fefcd065c.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)奥古斯丁·路易斯·柯西(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8256dc6189acb4b7725a77a51d14444f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eed0c5f79940f13c180beb9c6a6f08a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c1e6722de0b51ea234a160c8330edc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e258ab9e600435b37465092243d99f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abcdb7671e6056d842cdc88fb523467f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a1e005baa1be0da8cdf9173a9a46bd1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b73a89dbaa50fe6abfbae19c8cc57d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4c1254b9aeec2bbd01d0eecca66d708.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb5ba135022def1bcc1cddea66496706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ebbd1d0e4d44a11d9b0d65e73eef212.png)
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名校
2 . (1)已知
,求
的最大值.
(2)已知
且
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69ee6696dee035519e1ba7fb78269830.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e53b0009e24d67b65d24becf24e5bf2d.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bdbff19e3eead8ab58044a0aea36ad1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b4d4479034847ff0664c91be6f0fc98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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名校
3 . (1)求证:
,并指出等号何时成立;
(2)利用(1)的结论,试求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0638d7fa91f9d2a24fee94c539dadfd8.png)
(2)利用(1)的结论,试求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93b15c3db0f189fd2d0711fde5d761aa.png)
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名校
4 . “柯西不等式”是由数学家柯西在研究数学分析中的“流数”问题时得到的,但从历史的角度讲,该不等式应当称为柯西﹣﹣布尼亚科夫斯基﹣﹣施瓦茨不等式,因为正是后两位数学家彼此独立地在积分学中推而广之,才将这一不等式推广到完善的地步,在高中数学选修教材4﹣5中给出了二维形式的柯西不等式:(a2+b2)(c2+d2)≥(ac+bd)2当且仅当ad=bc(即
)时等号成立.该不等式在数学中证明不等式和求函数最值等方面都有广泛的应用.根据柯西不等式可知函数
的最大值及取得最大值时x的值分别为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aceadccb5f1527c79b0db952f17d8c52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04062599d9bdaf511eff078243956eae.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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2019-06-21更新
|
1097次组卷
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9卷引用:安徽省阜阳市临泉县第一中学2019-2020学年高一上学期12月月考数学试题