已知集合A={(x,y)|y≥|x-a|},B={(x,y)|y≤-a|x|+2a}(a≥0).
(1)證明A∩B≠∅;
(2)當(dāng)0≤a≤4時(shí),求由A∩B中點(diǎn)組成圖形面積的最大值.
【答案】
分析:(1)根據(jù)(0,a)∈A,(0,2a)∈B,可得A∩B≠∅.
(2)分類討論:當(dāng)2≤a≤4時(shí),A∩B中點(diǎn)組成三角形,當(dāng)0<a<2時(shí),A∩B中點(diǎn)組成四邊形,求出相應(yīng)的面積,利用導(dǎo)數(shù)求得面積的最大值,從而可得結(jié)論.
解答:![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/images0.png)
(1)證明:顯然(0,a)∈A.
當(dāng)x=0時(shí),y≤-a|x|+2a=2a,
∴(0,2a)∈B.∴A∩B≠∅.
(2)解:如左上圖,當(dāng)2≤a≤4時(shí),A∩B中點(diǎn)組成如圖所示△EFD,
∴E(0,2a)、F(-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/0.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/1.png)
)、D(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/2.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/3.png)
)、G(0,a).
∴S
△EFD=S
△EFG+S
△FGD=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/4.png)
a•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/5.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/6.png)
a•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/7.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/8.png)
.
當(dāng)0<a<2時(shí),A∩B中點(diǎn)組成如右上圖所示四邊形EFGH.
∴E(0,2a)、F(-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/9.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/10.png)
)、G(a,0)、H(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/11.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/12.png)
)、D(-2,0)、Q(2,0),
∴S
四邊形EFGH=S
△DEQ-S
△DFG-S
△GHQ=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/13.png)
×4×2a-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/14.png)
(a+2)•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/15.png)
-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/16.png)
(2-a)•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/17.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/18.png)
.
當(dāng)a=0時(shí),A∩B={(0,0)},顯然適合上式,
∴S=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/19.png)
當(dāng)0≤a<2時(shí),S=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/20.png)
,∴S′=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/21.png)
>0
∴S=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/22.png)
在[0,2)上是增函數(shù).∴0≤S<
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/23.png)
.
當(dāng)a≥2時(shí),S=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/24.png)
,∴S′=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/25.png)
>0,
∴S=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/26.png)
在[2,4]上是增函數(shù),∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/27.png)
≤S≤
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/28.png)
.
綜上所述,當(dāng)a=4時(shí),A∩B中點(diǎn)組成圖形面積取得最大值
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191526212899619/SYS201310241915262128996019_DA/29.png)
.
點(diǎn)評:本題考查A∩B中點(diǎn)組成圖形面積的計(jì)算,考查利用導(dǎo)數(shù)求最大值,考查分類討論的數(shù)學(xué)思想,屬于中檔題.