题目
题型:不详难度:来源:
已知函数f(x)=ax3+bx2-3x在x=±1处取得极值.
(1)求函数f(x)的解析式;
(2)求证:对于区间[-1,1]上任意两个自变量的值x1,x2,都有
|f(x1)-f(x2)|≤4;
(3)若过点A(1,m)(m≠-2)可作曲线y=f(x)的三条切线,求实数m的取值范围.
答案
即
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055509-16395.gif)
(II)∵f(x)=x3-3x,∴f′(x)=3x2-3=3(x+1)(x-1),
当-1<x<1时,f′(x)<0,故f(x)在区间[-1,1]上为减函数,
fmax(x)=f(-1)=2,fmin(x)=f(1)=-2
∵对于区间[-1,1]上任意两个自变量的值x1,x2,
都有|f(x1)-f(x2)|≤|fmax(x) -fmin(x)|
|f(x1)-f(x2)|≤|fmax(x)-fmin(x)|=2-(-2)=4
(III)f′(x)=3x2-3=3(x+1)(x-1),∵曲线方程为y=x3-3x,∴点A(1,m)不在曲线上.
设切点为M(x0,y0),则点M的坐标满足
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055509-93886.gif)
因
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055509-22659.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055509-37634.gif)
整理得
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055510-64604.gif)
∴关于x0方程
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055510-56369.gif)
设g(x0)=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055510-56369.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055510-77406.gif)
∴g(x0)在(-∞,0),(1,+∞)上单调递增,在(0,1)上单调递减.
∴函数g(x0)=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055510-56369.gif)
∴关于x0方程
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055510-56369.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055510-35214.gif)
故所求的实数a的取值范围是-3<m<-2.
解析
核心考点
试题【(本小题满分l4分)已知函数f(x)=ax3+bx2-3x在x=±1处取得极值.(1)求函数f(x)的解析式; (2)求证:对于区间[-1,1]上任意两个自变】;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/20191018/20191018055458-84468.gif)
(Ⅰ)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055458-19321.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055458-66815.gif)
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055459-83540.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055459-62616.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055459-68482.gif)
已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055445-85785.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055445-80774.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055445-21638.gif)
(Ⅰ)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055445-72054.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055445-72807.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055446-44380.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055446-30775.gif)
(Ⅱ)求满足下列条件的所有实数对
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055446-49223.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055446-30775.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055446-50087.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055446-54633.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055445-72807.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055447-27947.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055447-65999.gif)
(Ⅲ)对满足(Ⅱ)的条件的一个实数对
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055446-49223.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055447-84829.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055447-40277.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055448-55959.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055448-30422.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055448-82692.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055448-28487.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055448-55959.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055446-50087.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055438-98939.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055438-21914.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055439-14310.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055439-98344.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055439-75436.gif)
A.![]() |
B.![]() |
C.![]() |
D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055427-63405.gif)
(Ⅰ)设
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055427-92108.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055427-12911.gif)
(Ⅱ)若对任意
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055427-50953.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055428-46363.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055428-69122.gif)
设函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055417-55893.gif)
⑴求
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055418-50057.gif)
⑵若对任意的
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055418-18712.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055418-30299.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055418-24705.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055418-21999.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018055419-53718.gif)
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