【答案】
分析:(1)通過舉反例說明當(dāng)a=1時(shí),f(x)非奇非偶.
(2)利用絕對(duì)值的意義分段討論去掉絕對(duì)值符號(hào)將f(x)轉(zhuǎn)化為分段函數(shù);分別通過導(dǎo)數(shù)求兩段的最小值;比較兩段的最小值,挑出最小值為f(x)d的最小值.
解答:解:(1)當(dāng)a=1時(shí),f(x)=x
3-3|x-1|,(2分)
此時(shí)f(1)=1,f(-1)=-7,f(-1)≠f(1),f(-1)≠-f(1),∴f(x)是非奇非偶函數(shù).(5分)
(2)當(dāng)0≤x<1時(shí),f(x)=x
3-3a(1-x)=x
3+3ax-3a,
當(dāng)x≥1時(shí),f(x)=x
3-3a(x-1)=x
3-3ax+3a
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/0.png)
,(7分)
(i)當(dāng)0≤x<1時(shí),f'(x)=3x
2+3a,由于a>0,故f'(x)>0,∴f(x)在[0,1)內(nèi)單調(diào)遞增,此時(shí)[f(x)]
min=f(0)=-3a(9分)
(ii)當(dāng)x≥1時(shí),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/1.png)
,
令f'(x)=0,可得兩極值點(diǎn)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/2.png)
或
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/3.png)
,
①若0<a≤1,則
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/4.png)
,可得f(x)在[1,+∞)內(nèi)單調(diào)遞增,
結(jié)合(i)、(ii)可得此時(shí)[f(x)]
min=f(0)=-3a(11分)
②若a>1,則
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/5.png)
,可得f(x)在
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/6.png)
內(nèi)單調(diào)遞減,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/7.png)
內(nèi)單調(diào)遞增,
∴f(x)在[1,+∞)內(nèi)有極小值
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/8.png)
,
此時(shí)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/9.png)
而
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/10.png)
可得1<a≤9時(shí),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/11.png)
,a>9時(shí),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/12.png)
(14分)
∴綜合①②可得,當(dāng)0<a≤9時(shí),[f(x)]
min=f(0)=-3a,
當(dāng)a>9時(shí),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181018263045107/SYS201310241810182630451020_DA/13.png)
(15分)
點(diǎn)評(píng):本題考查通過舉反例說明一個(gè)命題不成立的方法、考查通過絕對(duì)值的意義去絕對(duì)值符號(hào)、考查分段函數(shù)的最值分段求,比較出各段的最值、考查利用導(dǎo)數(shù)求函數(shù)的最值、考查分類討論的數(shù)學(xué)思想方法.