题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101754-90170.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101754-81901.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101755-11603.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101755-17101.png)
(1)求实数
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101755-70253.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101755-64561.png)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101756-54762.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101756-59296.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101756-95424.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101756-65319.png)
(3)若正实数
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101757-52343.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101757-80811.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101757-34634.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101757-59358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101757-92578.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101758-45268.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101758-24684.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101758-49229.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101758-30047.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101759-12283.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101759-86638.png)
解析
试题分析:(1)由
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101754-90170.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101754-81901.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101755-11603.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101755-17101.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101759-37181.png)
(2)对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101756-59296.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101756-95424.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101800-41989.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101800-53824.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101800-89450.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101756-54762.png)
(3)若正实数
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101757-52343.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101757-80811.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101757-34634.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101757-59358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101758-45268.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101758-24684.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101758-49229.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101758-30047.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101759-12283.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101801-70677.png)
(1)∵
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101801-75401.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101801-17620.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101759-37181.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101801-24486.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101802-62415.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101802-69272.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101802-14657.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101755-64561.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101802-44317.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101802-70876.png)
(2)由
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101756-95424.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101803-21758.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101803-14571.png)
令函数
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101800-41989.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101800-53824.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101800-89450.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101803-35437.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101803-66520.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101800-89450.png)
而
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101804-63867.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101804-30862.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101759-86638.png)
(3)证明:不妨设
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101804-31840.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101757-80811.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101804-16211.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101805-11674.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101805-56474.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101805-65569.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101805-94252.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101806-91743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101806-80254.png)
其中
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101806-43482.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101806-21790.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101807-82537.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101807-30121.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101807-48972.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101807-24527.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101807-73945.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101808-38422.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101808-85204.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101802-44317.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101808-49347.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101808-46043.png)
从而有
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101757-59358.png)
该不等式能更进一步推广:
已知
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101809-99691.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101758-49229.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101757-92578.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101758-45268.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101758-30047.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101759-12283.png)
下面用数学归纳法加以证明:
i)当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101809-11763.png)
ii)假设当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101809-15316.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101809-14348.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101810-46080.png)
则当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101810-76192.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101810-26106.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101801-70677.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101810-50651.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101811-11630.png)
在该不等式的两边同时乘以正数
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101811-82627.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101811-14519.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101811-32493.png)
在此不等式的两边同时加上
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101811-14467.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101812-20730.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101812-34167.png)
该不等式的左边再利用i)的结论可得:
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101812-60941.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101813-76299.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101813-28420.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101813-52684.png)
所以,当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101810-76192.png)
综上,对
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101813-56554.png)
核心考点
试题【设(是自然对数的底数,),且.(1)求实数的值,并求函数的单调区间;(2)设,对任意,恒有成立.求实数的取值范围;(3)若正实数满足,,试证明:;并进一步判断:】;主要考察你对函数的相关概念等知识点的理解。[详细]
举一反三
A.x-1 | B.x+1 | C.2x+1 | D.3x+3 |
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101715-15623.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101716-78056.png)
A.1 | B.3 | C.15 | D.30 |
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101646-87329.png)
A.(-3,1)∪(3,+∞) | B.(-3,1)∪(2,+∞) |
C.(-1,1)∪(3,+∞) | D.(-∞,-3)∪(1,3) |
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101614-51182.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101614-51182.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204101614-63181.png)
A.(0,![]() | B.(![]() ![]() |
C.(![]() ![]() | D.[0,![]() |
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