题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052356-34039.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052356-73452.png)
(1)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052356-72624.png)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052356-34452.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052356-13695.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052357-46971.png)
(3)斜率为
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052357-77453.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052357-89564.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052357-17110.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052357-49048.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052357-90988.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052357-39430.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052357-61532.png)
当a<0时,F(x)在
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052358-87959.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052358-43676.png)
解析
试题分析:(1)f"(x)=lnx+1(x>0),令f"(x)=0,得
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052358-62758.png)
∵当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052358-10081.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052358-69635.png)
f"(x)>0,
∴当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052358-76707.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052357-61532.png)
(2)F(x)=ax2+lnx+1(x>0),
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052358-30922.png)
①当a≥0时,恒有F"(x)>0,F(x)在(0,+∞)上是增函数;
②当a<0时,
令F"(x)>0,得2ax2+1>0,解得
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052359-77361.png)
令F"(x)<0,得2ax2+1<0,解得
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052359-74589.png)
综上,当a≥0时,F(x)在(0,+∞)上是增函数;
当a<0时,F(x)在
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052358-87959.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052358-43676.png)
(3)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052359-58902.png)
要证
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052359-31730.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052359-93146.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052359-59769.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052400-23229.png)
则只要证
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052400-98206.png)
故等价于证lnt<t﹣1<tlnt(t>1)(*).
①设g(t)=t﹣1﹣lnt(t≥1),则
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052400-81004.png)
故g(t)在[1,+∞)上是增函数,
∴当t>1时,g(t)=t﹣1﹣lnt>g(1)=0,即t﹣1>lnt(t>1).
②设h(t)=tlnt﹣(t﹣1)(t≥1),则h"(t)=lnt≥0(t≥1),故h(t)在[1,+∞)上是增函数,
∴当t>1时,h(t)=tlnt﹣(t﹣1)>h(1)=0,即t﹣1<tlnt(t>1).
由①②知(*)成立,得证. 12分
点评:导数本身是个解决问题的工具,是高考必考内容之一,高考往往结合函数甚至是实际问题考查导数的应用,求单调、最值、完成证明等,请注意归纳常规方法和常见注意点
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052339-34177.png)
(1)若函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052339-31687.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052340-90423.png)
①求实数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052340-82379.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052340-41726.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052340-12130.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052341-21661.png)
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052341-74587.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052341-27522.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052341-36454.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052341-52113.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052331-26961.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052331-47106.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052331-76862.png)
(Ⅰ)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052331-47106.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052332-22480.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052332-69772.png)
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052332-69772.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052332-31750.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052321-14593.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052321-61856.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052321-73021.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052321-50547.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052322-35460.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052322-72415.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052312-86502.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052312-30367.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052312-65405.png)
(1)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052312-69354.png)
(2)已知
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052312-12804.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052312-26548.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052312-98209.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052312-10413.png)
(3)设
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052313-40242.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052313-74347.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052313-53000.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052301-53755.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018052301-90251.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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