题目
题型:陕西省模拟题难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004140-63842.png)
(1)求函数f(x)的单调区间;
(2)记函数y=F(x)的图象为曲线C。设点A(x1,y1),B(x2,y2)是曲线C上的不同两点。如果在曲线C上存在点M(x0,y0),使得:①
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004140-29012.png)
答案
由已知得,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004141-35273.png)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004141-73984.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004141-84874.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004141-93866.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004142-94615.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004142-60624.png)
所以函数f(x)在(0,1)上单调递增,在(1,+∞)上单调递减;
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004142-10728.png)
①当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004142-13635.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004142-62242.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004143-18412.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004143-64253.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004143-81474.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004143-79574.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004144-17205.png)
所以,函数f(x)在
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004144-92694.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004144-59272.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004144-52451.png)
②当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004144-97117.png)
③当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004145-53427.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004145-26705.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004145-39752.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004145-67152.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004145-76001.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004146-37217.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004146-30441.png)
所以,函数f(x)在(0,1)和
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004146-78983.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004146-52293.png)
综上所述,(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004147-75477.png)
在(1,+∞)上单调递减;
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004147-97207.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004147-16136.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004147-94490.png)
(3)当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004147-32481.png)
(4)当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004148-80121.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004148-79365.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004148-26348.png)
(Ⅱ)假设函数f(x)存在“中值相依切线”,
设
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004148-74736.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004149-39990.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004149-27076.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004149-78371.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004149-92233.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004149-35488.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004150-42625.png)
曲线在点
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004150-73707.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004150-15063.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004150-49830.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004151-66401.png)
依题意得:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004151-64336.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004151-92282.png)
化简可得:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004151-63518.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004152-17763.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004152-18987.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004152-93271.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004152-74916.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004152-84989.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004153-30489.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004153-95357.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004153-63972.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004153-72782.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004154-70835.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004154-61180.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004154-78918.png)
显然有g(t)>2恒成立;
所以在(1,+∞)内不存在t,使得
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004154-65326.png)
综上所述,假设不成立,所以,函数f(x)不存在“中值相依切线”。
核心考点
试题【已知函数f(x)=lnx-ax2+(a-1)x(a∈R且a≠0),(1)求函数f(x)的单调区间;(2)记函数y=F(x)的图象为曲线C。设点A(x1,y1),】;主要考察你对函数的单调性与导数等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004130-94513.png)
(1)若f(x)在x=1处取得极值,求f(x)的单调区间;
(2)若m=2,令h(x)=f(x)-3x,证明:对任意的x1,x2∈[1,2],恒有|h(x1)-h(x2)|<1。
(Ⅰ)求b,c的值;
(Ⅱ)求函数f(x)的单调区间;
(Ⅲ)若方程f(x)=m恰有两个不等的实根,求m的取值范围。
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004112-50578.gif)
(1)若
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004112-65608.gif)
(2)求证:当0<a≤2时,f(x)在
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004112-85912.gif)
(3)若对任意的a∈(1,2),总存在
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004113-11684.gif)
(Ⅰ)要使y=f(x)在(0,1)上单调递增,求a的取值范围;
(Ⅱ)当a>0时,若函数满足y极小值=1,y极大值=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004058-10276.png)
(Ⅲ)若x∈[0,1]时,y=f(x)图象上任意一点处的切线倾斜角为θ,求当0≤θ≤
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004059-30809.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004048-38907.gif)
(Ⅰ)若f(x)在x=1处取得极值,求a的值;
(Ⅱ)求f(x)的单调区间;
(Ⅲ)若f(x)的最小值为1,求a的取值范围。
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