名校
1 . 英国数学家泰勒发现了如下公式:
,其中
,此公式有广泛的用途,例如利用公式得到一些不等式:当
时,
,
.
(1)证明:当
时,
;
(2)设
,若区间
满足当
定义域为
时,值域也为
,则称为
的“和谐区间”.
(i)
时,
是否存在“和谐区间”?若存在,求出
的所有“和谐区间”,若不存在,请说明理由;
(ii)
时,
是否存在“和谐区间”?若存在,求出
的所有“和谐区间”,若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9c8d6b7790572ee26dac80e0c7fe648.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b80c875ad8fafc41d5c82baf23bb5e4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4138f6987cd2ee9e56b2ac80e84f9e24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ee051a4daa81ab32ef9c153ecf90e02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/305249d05ecc23ee86ae55f7bf8566e1.png)
(1)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4138f6987cd2ee9e56b2ac80e84f9e24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9f80e45170c557aed6187a6bd11177d.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f95d2a9ba5f50d14cdee5ecda28461a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.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/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(i)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf0086b054ef120408acac806a1b1318.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(ii)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0db2c49919467a2e14540f2aabd05cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
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2022-02-22更新
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