已知函數(shù)f(x)=x|x-a|+2x.
(1)若函數(shù)f(x)在R上是增函數(shù),求實數(shù)a的取值范圍;
(2)求所有的實數(shù)a,使得對任意x∈[1,2]時,函數(shù)f(x)的圖象恒在函數(shù)g(x)=2x+1圖象的下方;
(3)若存在a∈[-4,4],使得關于x的方程f(x)=tf(a)有三個不相等的實數(shù)根,求實數(shù)t的取值范圍.
【答案】
分析:(1)由題意知f(x)在R上是增函數(shù),則
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/0.png)
即-2≤a≤2,則a范圍.
(2)由題意得對任意的實數(shù)x∈[1,2],f(x)<g(x)恒成立,即
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/1.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/2.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/3.png)
,故只要
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/4.png)
且
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/5.png)
在x∈[1,2]上恒成立即可,在x∈[1,2]時,只要
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/6.png)
的最大值小于a且
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/7.png)
的最小值大于a即可.由此可知答案.
(3)當-2≤a≤2時,f(x)在R上是增函數(shù),則關于x的方程f(x)=tf(a)不可能有三個不等的實數(shù)根存在a∈(2,4],方程f(x)=tf(a)=2ta有三個不相等的實根,則
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/8.png)
,即存在a∈(2,4],使得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/9.png)
即可,由此可證出實數(shù)t的取值范圍為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/10.png)
.
解答:解:(1)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/11.png)
由f(x)在R上是增函數(shù),則
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/12.png)
即-2≤a≤2,則a范圍為-2≤a≤2;(4分)
(2)由題意得對任意的實數(shù)x∈[1,2],f(x)<g(x)恒成立,
即x|x-a|<1,當x∈[1,2]恒成立,即
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/13.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/14.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/15.png)
,故只要
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/16.png)
且
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/17.png)
在x∈[1,2]上恒成立即可,
在x∈[1,2]時,只要
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/18.png)
的最大值小于a且
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/19.png)
的最小值大于a即可,(6分)
而當x∈[1,2]時,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/20.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/21.png)
為增函數(shù),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/22.png)
;
當x∈[1,2]時,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/23.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/24.png)
為增函數(shù),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/25.png)
,
所以
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/26.png)
;(10分)
(3)當-2≤a≤2時,f(x)在R上是增函數(shù),則關于x的方程f(x)=tf(a)不可能有三個不等的實數(shù)根;(11分)
則當a∈(2,4]時,由
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/27.png)
得x≥a時,f(x)=x
2+(2-a)x對稱軸
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/28.png)
,
則f(x)在x∈[a,+∞)為增函數(shù),此時f(x)的值域為[f(a),+∞)=[2a,+∞),x<a時,f(x)=-x
2+(2+a)x對稱軸
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/29.png)
,
則f(x)在
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/30.png)
為增函數(shù),此時f(x)的值域為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/31.png)
,f(x)在
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/32.png)
為減函數(shù),此時f(x)的值域為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/33.png)
;
由存在a∈(2,4],方程f(x)=tf(a)=2ta有三個不相等的實根,則
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/34.png)
,
即存在a∈(2,4],使得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/35.png)
即可,令
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/36.png)
,
只要使t<(g(a))
max即可,而g(a)在a∈(2,4]上是增函數(shù),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/37.png)
,
故實數(shù)t的取值范圍為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/38.png)
;(15分)
同理可求當a∈[-4,-2)時,t的取值范圍為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/39.png)
;
綜上所述,實數(shù)t的取值范圍為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181230578587619/SYS201310241812305785876019_DA/40.png)
.(16分)
點評:本題考查函數(shù)性質(zhì)的綜合應用,解題時要認真審題.