已知f(x)=2x可以表示成一個奇函數(shù)g(x)與一個偶函數(shù)h(x)之和,若關(guān)于x的不等式ag(x)+h(2x)≥0對于x∈[1,2]恒成立,則實數(shù)a的最小值是 .
【答案】
分析:由題意可得g(x)+h(x)=2
x①,g(-x)+h(-x)=-g(x)+h(x)=2
-x②從而可得h(x)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/0.png)
,g(x)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/1.png)
而ag(x)+h(2x)≥0對于x∈[1,2]恒成立即a
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/2.png)
對于x∈[1,2]恒成立即
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/3.png)
對于x∈[1,2]恒成立,只要求出函數(shù)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/4.png)
的最大值即可
解答:解:f(x)=2
x可以表示成一個奇函數(shù)g(x)與一個偶函數(shù)h(x)之和
∴g(x)+h(x)=2
x①,g(-x)+h(-x)=-g(x)+h(x)=2
-x②
①②聯(lián)立可得,h(x)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/5.png)
,g(x)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/6.png)
ag(x)+h(2x)≥0對于x∈[1,2]恒成立
a
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/7.png)
對于x∈[1,2]恒成立
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/8.png)
對于x∈[1,2]恒成立
t=2
x-2
-x,x∈[1,2],t∈
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/9.png)
則t
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/10.png)
在t∈
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/11.png)
單調(diào)遞增,
t=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/12.png)
時,則t
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/13.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/14.png)
a
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101223032298763949/SYS201311012230322987639012_DA/15.png)
故答案為:
點評:本題主要考查了奇偶函數(shù)的定義的應(yīng)用,函數(shù)的恒成立的問題,常會轉(zhuǎn)化為求函數(shù)的最值問題,體現(xiàn)了轉(zhuǎn)化思想的應(yīng)用.