解题方法
1 . 若函数
有且仅有一个极值点
,函数
有且仅有一个极值点
,且
,则称
与
具有性质
.
(1)函数
与
是否具有性质
?并说明理由.
(2)已知函数
与
具有性质
.
(i)求
的取值范围;
(ii)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d0112af71e654cc86c8d5056fdbb2a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/861433f0c552c5bef8d8c03682d91858.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7091d529281abff275ef19b9197445a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d0112af71e654cc86c8d5056fdbb2a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/861433f0c552c5bef8d8c03682d91858.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d77fbccb3ae7f7836d16dfb4952e4cc.png)
(1)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47d2a4a953934dcaf87f2ce64c6dab4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0482361cb97998139441fe0deb23578e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a9d58d2cef0d86281fdde5895a129a6.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b00477b17f5248a7301290d260a6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e5eef1c41c5ed94a4944e062bcfeeeb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f36ea881341f274252964bc3a9fc5693.png)
(i)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(ii)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/926521541bf1c18ac229afc5ec2d9b51.png)
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解题方法
2 . (1)证明:当
时,
;
(2)已知函数
,若
是
的极小值点,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca542e78b7d77d008c9c4752afa91a55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2acbfad5866c69207ec8ca7f39fe7c7d.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0172d1295992e244650b29b964afc052.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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3 . 已知函数
,则下列说法正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/514df790653cf281af6323d283f72acd.png)
A.若![]() ![]() ![]() |
B.若![]() ![]() |
C.存在![]() ![]() |
D.若![]() ![]() ![]() |
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解题方法
4 . 已知函数
.
(1)当
时,求曲线
在点
处的切线方程;
(2)若
恒成立,求
的取值范围;
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/536d662ffb82903b39d82d7053aadd2f.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/302337058242c7b78e3eb4ac7210b7ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db40d5295942e85ec07a3728c7ad308d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6b0c3cf87f438d1fe0f6fca51b40be2.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8e75f547129d14ffb61324fb58492ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8db9ea9575be1bdcd679099b47e45c23.png)
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名校
解题方法
5 . 已知函数
.
(1)若
的极小值为-4,求
的值;
(2)若
有两个不同的极值点
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ec485b17e78ee033856963013d5dec5.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e20a11121e7ccc795552da69bb921071.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b06e1fc0c843262c1463d9cf04bb835.png)
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名校
6 . 极值的广义定义如下:如果一个函数在一点的一个邻域(包含该点的开区间)内处处都有确定的值,而以该点处的值为最大(小),这函数在该点处的值就是一个极大(小)值.
对于函数
,设自变量x从
变化到
,当
,
是一个确定的值,则称函数
在点
处右可导;当
,
是一个确定的值,则称函数
在点
处左可导.当函数
在点
处既右可导也左可导且导数值相等,则称函数
在点
处可导.
(1)请举出一个例子,说明该函数在某点处不可导,但是该点是该函数的极值点;
(2)已知函数
.
(ⅰ)求函数
在
处的切线方程;
(ⅱ)若
为
的极小值点,求a的取值范围.
对于函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d47db6b322f19cefcd70d0f8433917f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1459527045d2d138d1ee4b441aade53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55e6c43466d743ea37287656b85821c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/570e972f51cd1d11acfd80a01c0512fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55e6c43466d743ea37287656b85821c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
(1)请举出一个例子,说明该函数在某点处不可导,但是该点是该函数的极值点;
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fae34b02b147bb49131a01316e07efaf.png)
(ⅰ)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d1735755a0f13e329df682712dd1d6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
(ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
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解题方法
7 . 已知函数
,其极大值点和极小值点分别为
,记点
,直线
交曲线
于点
,若存在常数
,使得
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3cb8d72bb2e281b943b3b430138ef7.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ef3e79110067a46276f0869bea25af5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2790aecde6df3b05f195ab8daddb56be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7e11699fa0813a5e9d6baf694f5bf4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/357924c44549675683398a0b7c9bcb26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3cb8d72bb2e281b943b3b430138ef7.png)
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解题方法
8 . 如果三个互不相同的函数
,
,
在区间
上恒有
或
,则称
为
与
在区间
上的“分割函数”.
(1)证明:函数
为函数
与
在
上的分割函数;
(2)若函数
为函数
与
在
上的“分割函数”,求实数
的取值范围;
(3)若
,且存在实数
,使得函数
为函数
与
在区间
上的“分割函数”,求
的最大值.
![](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/f786a5701dc1a8a015e8843c3360151b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d61157daf46974d1a08cd4b465a92abe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af28992f69a3dab36678839b8a5e5720.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f786a5701dc1a8a015e8843c3360151b.png)
![](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/8455657dde27aabe6adb7b188e031c11.png)
(1)证明:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38e00b49aa78de649f34d8bb9d5179ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8190660962c1d992d7d61a69c21a2737.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6583d861ba47d9123d75dc90b8df0c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7334919736e5ed881f691e4ca738b4ce.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8352b2e643a7ce605334f1b0e572bfb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a05ba1bc6c7bc24879b2a17ef2351c59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23afc43a8c5b8cfe6bf2a1caed920c01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2372f424431ce7b547a66b7d61d75421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a06e1578853d2072cef33395de8784d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3046b064d62833c805c84d5a8866c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8829770dd5a85c2ecaf82edea669869d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aea89f800e9af713ec91e00fb287008.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4ed21127710fb6adcf694bd14aff321.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57b85a97933a1d984f6e484b4021c800.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64da75a02173c2a5eb40f4c68d0f4f36.png)
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9 . 已知函数
,其中
表示
,
中的最大值,若函数
有3个零点,则实数
的取值范围是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07b532b68bfc793ca4826283d97458ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/385e1459ada24f66206d0de24a85ad0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18c49ca8562b98657ca9c499093f7233.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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10 . 已知
,下列四个结论:①
,②
,③
,④
.其中错误的个数是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e1bb486cbf98ae341a35a17e79fae7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7b0e6eb011614deef2d50ceab01c8d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7e0b26fa1a2c42bf24833e241ea25a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ec504e75b5fe6435a94047a2f0b9443.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9dd82388f903f53c58ff6d655dd7d14.png)
A.1 | B.2 | C.3 | D.4 |
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