题目
题型:不详难度:来源:
已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015812-49809.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015812-90532.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015812-82762.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015812-24035.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015812-85605.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015812-24035.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015812-33737.png)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015812-90532.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015812-30463.png)
(2)证明:当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015813-89161.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015813-11070.png)
(3)证明:对任意给定的正数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015813-72947.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015813-93103.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015813-77536.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-34415.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-61952.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-51178.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-65234.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-51178.png)
(2)见解析.(3)见解析.
解析
试题分析:(1)由
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-41540.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015815-45707.png)
从而
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015815-88024.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015815-75719.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-61952.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015815-59664.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015815-41078.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-51178.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015816-79186.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015816-17270.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-51178.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-61952.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-51178.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-65234.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-51178.png)
(2)令
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015816-79908.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015816-84980.png)
根据
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015817-72822.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015817-38390.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015817-38075.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015817-64110.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015817-71470.png)
(3)思路一:对任意给定的正数c,取
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015818-77885.png)
根据
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015818-54642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015818-54232.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015818-91920.png)
思路二:令
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015818-40247.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015819-77970.png)
分
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015819-22383.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015819-29408.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015819-79040.png)
思路三:就①
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015820-85331.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015820-22512.png)
试题解析:解法一:
(1)由
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015820-56577.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015820-19746.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-41540.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015815-45707.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015820-74465.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015815-88024.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015815-75719.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-61952.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015815-59664.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015815-41078.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-51178.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015816-79186.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015816-17270.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-51178.png)
所以当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-61952.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-51178.png)
且极小值为
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-97755.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015814-51178.png)
(2)令
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015816-79908.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015816-84980.png)
由(1)得,
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015817-72822.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-74981.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015817-38390.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015817-38075.png)
所以当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015817-64110.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015817-71470.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015818-54642.png)
(3)对任意给定的正数c,取
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015818-77885.png)
由(2)知,当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015817-64110.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015818-54642.png)
所以当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015818-54232.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015818-91920.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-16005.png)
因此,对任意给定的正数c,总存在
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-43463.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-63005.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-16005.png)
解法二:(1)同解法一.
(2)同解法一.
(3)令
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015818-40247.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-16005.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015822-81724.png)
而要使
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015822-81724.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015822-93128.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015819-77970.png)
①若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015819-22383.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015822-94083.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015817-64110.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015822-60911.png)
即对任意
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015823-10398.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015823-73992.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-63005.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-16005.png)
②若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015819-29408.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015819-79040.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015823-45891.png)
所以当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015823-73220.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015823-12382.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015824-73966.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015824-37330.png)
取
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015824-73852.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015824-51851.png)
易知
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015824-81973.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015825-28085.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015825-43772.png)
因此对任意
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015825-22220.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015825-96055.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-63005.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-16005.png)
综上,对任意给定的正数c,总存在
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-43463.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-63005.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-16005.png)
解法三:(1)同解法一.
(2)同解法一.
(3)①若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015820-85331.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015823-73992.png)
由(2)的证明过程知,
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015825-56156.png)
所以当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-63005.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015826-41629.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-16005.png)
②若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015820-22512.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015826-75096.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015826-41487.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015826-54142.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015826-31673.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015827-51234.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015823-12382.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015824-73966.png)
取
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015827-28607.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015827-29429.png)
易知
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015827-26215.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015824-73966.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015827-34613.png)
所以当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-63005.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015828-62893.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-16005.png)
综上,对任意给定的正数c,总存在
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-43463.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-63005.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015821-16005.png)
核心考点
试题【(本小题满分14分)已知函数(为常数)的图像与轴交于点,曲线在点处的切线斜率为.(1)求的值及函数的极值;(2)证明:当时,(3)证明:对任意给定的正数,总存在】;主要考察你对常见函数的导数等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015807-34779.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015807-65486.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015803-11861.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015803-76426.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015743-66405.png)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015743-29366.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015744-48790.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015744-14029.png)
(2)讨论函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015744-61298.png)
(3)若对任意
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015744-83315.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015744-62800.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015731-18334.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015731-68167.png)
求b;若存在
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015731-68327.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015732-65956.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015707-94964.png)
(1当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015708-48667.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015708-45862.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015708-10539.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015708-57336.png)
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015708-89282.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015709-93021.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015709-96847.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015709-79329.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015709-54692.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015710-39410.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018015710-29141.png)
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