题目
题型:不详难度:来源:
(1)求常数a的值;(2)若存在x使不等式
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045639-57161.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045639-94830.png)
(3)对于函数y=f(x)和y=g(x)公共定义域内的任意实数x0,我们把|f(x0)-g(x0)|的值称为两函数在x0处的偏差.求证:函数y=f(x)和y=g(x)在其公共定义域内的所有偏差都大于2.
答案
解析
试题分析:(1)求出交点,切线平行即导数值相等可解;(2)转化为新函数,求出导数,利用单调性极值解;(3)构造新函数求导,利用单调性证明.
试题解析:(1)f(x)与坐标轴的交点为(0,a),f′(0)=a,g(x)与坐标轴的交点为(a,0),g′(a)=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045639-72135.png)
∴a=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045639-72135.png)
(2
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045639-57161.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045639-94830.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045639-94830.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045639-94830.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045639-44111.png)
∵x>0,∴
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045640-73589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045639-94830.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045640-38444.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045640-42546.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045640-80313.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045640-73589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045639-94830.png)
∴h(x)在(0,+∞)上是减函数,因此h(x)<h(0)=0. ∴实数m的取值范围是(-∞,0).
(3)y=f(x)与y=g(x)的公共定义域为(0,+∞),|f(x)-g(x)|=|ex-lnx|=ex-lnx.
令h(x)=ex-x-1,则h′(x)=ex-1>0.∴h(x)在(0,+∞)上是增函数.
故h(x)>h(0)=0,即ex-1>x. ①
令m(x)=lnx-x+1,则m′(x)=
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045641-44028.png)
当x>1时,m′(x)<0,当0<x<1时,m′(x)>0.∴m(x)有最大值m(1)=0,因此lnx+1<x. ②
由①②,得ex-1>lnx+1,即ex-lnx>2.
∴函数y=f(x)和y=g(x)在其公共定义域内的所有偏差都大于2.
核心考点
试题【已知函数f(x)=aex,g(x)=lnx-lna,其中a为常数,e=2.718…,且函数y=f(x)和y=g(x)的图像在它们与坐标轴交点处的切线互相平行.(】;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/20191018/20191018045629-18115.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045629-94531.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045630-58789.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045630-48602.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045630-64156.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045630-50364.png)
A.11 | B.10 | C.9 | D.8 |
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045611-35103.png)
(Ⅰ)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045612-82539.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045612-25916.png)
(Ⅱ)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045612-13396.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045612-18925.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045613-46358.png)
(Ⅲ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045613-86928.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045613-53137.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045555-43051.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045556-64409.png)
(1)若曲线
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045556-66486.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045556-57265.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045556-21078.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045557-63054.png)
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045557-79240.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045557-49340.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045557-90218.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045557-70094.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045558-22879.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045558-95897.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045558-91477.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045542-61960.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045542-12251.png)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045543-70494.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045543-64422.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045543-64422.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045543-36858.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045543-95251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045543-66094.png)
(3)若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045544-81533.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045544-16720.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045543-66094.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045531-93485.png)
(1)若函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045531-76572.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045531-90428.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045532-81223.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045532-98432.png)
(2)在(1)的条件下,求函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045531-76572.png)
(3)若函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045532-83139.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045532-70693.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018045532-98432.png)
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