题目
题型:河南省模拟题难度:来源:
(1)求函数f(x)的最小值;
(2)设F(x)=a
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073840-37357.png)
(3)斜率为k的直线与曲线y=f"(x)交于A(
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073840-18494.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073841-58780.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073841-66767.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073841-89214.png)
答案
令f"(x)=0,得
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073841-44388.png)
∵当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073841-65604.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073841-63449.png)
∴当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073841-13854.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073841-92918.png)
(2)F(x)=a
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073841-49725.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073842-39966.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2a
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073842-50080.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073842-66468.png)
令F"(x)<0,得2a
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073842-41306.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073842-80580.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/20191019/20191019073842-18680.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073842-96111.png)
(3)证:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073843-52197.png)
要证
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073843-20791.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073843-76440.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073843-40275.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073843-84158.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073844-54457.png)
由t>1知lnt>0,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/20191019/20191019073844-66377.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.
核心考点
试题【设函数f(x)=xlnx(x>0).(1)求函数f(x)的最小值;(2)设F(x)=a+f"(x)(a∈R),讨论函数F(x)的单调性;(3)斜率为k的直线与曲】;主要考察你对函数极值与最值等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073826-65694.png)
(I)求函数f(x)的单调区间和极值;
(II)若
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073826-73848.png)
(1)若函数f(x)在x=1处与直线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073819-19507.png)
①求实数a,b的值;
②求函数
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073820-78854.png)
(2)当b=0时,若不等式f(x)≥m+x对所有的
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073820-23052.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073807-32014.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073807-84526.png)
(1)当a=b=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073807-55930.png)
(2)令F(x)=f(x)+
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073807-98703.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073808-62473.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073808-33721.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073808-14832.png)
(3)当a=0,b=﹣1时,方程2mf(x)=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073808-86354.png)
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