1 . 定义区间
的长度均为
,其中
.
(1)若函数
的定义域为
值域为
,写出区间长度
的最大值;
(2)已知
,求证:关于
的不等式
的解集构成的各区间的长度和为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36051bfe736319addd75dbad8363ace4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08d9560a5a060961e957ffeff10a29aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ecebb31ea8ee6db240b42a5ee2c5e88.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/927adc3ef749d09e76374ffbd7c50da9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd75bc1dd7b2c6785a5a913827671d36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7569cd7e9b31ad838230133b9bc8314.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a929905a2bcccbad3a43e86f05dbdf3.png)
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2 . 已知函数
,其中
,设函数
的反函数为
.
(1)记函数
的导函数为
,函数
的导函数为
,若存在
满足
,证明:
;
(2)若函数
与函数
的图象有两个交点,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d76ee3b131ecd6aa1aacf7fb7b3eb15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d33da711e50e96568facb18cef27165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(1)记函数
![](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/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22add663bd26e87d972a10dc5fd9ada1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c62d347a6df5562e43a4e148964ceb86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d77889d413e5957d03560c366da4866.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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解题方法
3 . 已知函数
的图象过原点.
(1)当
时,求该函数的解析式,判断并证明其奇偶性;
(2)若该函数图象无限接近直线
但又不与该直线相交.
①求
和
的值;
②请画出该函数图象,并写出其单调区间(不必证明).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fec27becd5bcdcfa14f988539555d833.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf0086b054ef120408acac806a1b1318.png)
(2)若该函数图象无限接近直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9355031ea0b2dc9cef3777621bc6d38.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
②请画出该函数图象,并写出其单调区间(不必证明).
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