名校
解题方法
1 . 若集合A具有以下性质,则称集合A是“好集”:①
;②若
,则
,且
时,
.
(1)分别判断集合
,有理数集
是否是“好集”,并说明理由;
(2)设集合
是“好集”,求证:若
,则
;
(3)对任意的一个“好集”A,判断下面命题的真假,并说明理由;命题:若
,则必有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6c9b39503b6484104862e21772b1431.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e03cb9923332c1afa835e98fa24e2f27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c46de01c5104b9112a688df37eadb000.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38f0e9c04402a0ffdaa25c3e3c82c7dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cd77104cc745d1e0e262122da34482d.png)
(1)分别判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a05551b1d4b65f27a932c33ddb1cb6ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
(2)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e03cb9923332c1afa835e98fa24e2f27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/957d41dbe52b49c3a7339e3519a3fe84.png)
(3)对任意的一个“好集”A,判断下面命题的真假,并说明理由;命题:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e03cb9923332c1afa835e98fa24e2f27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ae4f0ccdfc1206d809e581449d0452e.png)
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名校
2 . 设函数
,其中
,若任意
均有
,则称函数
是函数
的控制函数”,且对于所有满足条件的函数
在
处取得的最小值记为
.
(1)若
,试问
是否为
的控制函数”;
(2)若
,使得直线
是曲线
在
处的切线,证明:函数
为函数
的控制函数,并求“
”的值;
(3)若曲线
在
处的切线过点
,且
,证明:当且仅当
或
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11d0d9cf90ee9e4216f6c5e19f7f4d18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cacd894a237683d42c389bfa5c27936c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1376168658dbe7f5b7f4d75fb1db545a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b4955c5adc717b7f6f0b975e0724ff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecea80c2b9483e2c65d7572598a48dbd.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d709d206efc9c004cf7ba5301aad67e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4d795709b0abcf47bceec2250f2f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e915b67f8f747698b8b46d37bc453667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94376e3e25de7fa4e506d40446b22ffc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e915b67f8f747698b8b46d37bc453667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55679c4d0d7c781f5db02eedb98baa4b.png)
(3)若曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0fa12e23f7017e424166ba4622b0e99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53a948d2f7732d7f03e986c63712089b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2023d0f4982eec32fae3b57bec6d8e31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/436b2649162a1b61b6ef0ab2bda35bc4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4580cc037c0c760c728cdbb74a8154c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9de7f7734539f4ceb08561cd4d1ecbc6.png)
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2023-01-08更新
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