已知函數(shù)f(x)=x2-x+alnx
(1)當(dāng)x≥1時,f(x)≤x2恒成立,求a的取值范圍;
(2)討論f(x)在定義域上的單調(diào)性.
【答案】
分析:(1)先利用參數(shù)分離法將a分離出來,然后研究函數(shù)的最值,使參數(shù)a恒小于函數(shù)的最小值即可;
(2)先確定函數(shù)的定義域然后求導(dǎo)數(shù)fˊ(x),在函數(shù)的定義域內(nèi)解不等式fˊ(x)>0和fˊ(x)<0,主要進(jìn)行分離討論.
解答:解:(1)由f(x)≤x
2恒成立,得:alnx≤x在x≥1時恒成立
當(dāng)x=1時a∈R(2分)
當(dāng)x>1時即
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/0.png)
,令
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/1.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/2.png)
(4分)
x≥e時g'(x)≥0,g(x)在x>e時為增函數(shù),g(x)在x<e時為減函數(shù)
∴g
min(x)=e∴a≤e(6分)
(2)解:f(x)=x
2-x+alnx,f′(x)=2x-1+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/3.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/4.png)
,x>0
(1)當(dāng)△=1-8a≤0,a≥
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/5.png)
時,f′(x)≥0恒成立,
f(x)在(0,+∞)上為增函數(shù).(8分)
(2)當(dāng)a<
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/6.png)
時
①當(dāng)0<a<
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/7.png)
時,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/8.png)
,
f(x)在
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/9.png)
上為減函數(shù),
f(x)在
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/10.png)
上為增函數(shù).(11分)
②當(dāng)a=0時,f(x)在(0,1]上為減函數(shù),f(x)在[1,+∞)上為增函數(shù)(12分)
③當(dāng)a<0時,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/11.png)
,故f(x)在(0,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/12.png)
]上為減函數(shù),
f(x)在[
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182207330674440/SYS201310241822073306744015_DA/13.png)
,+∞)上為增函數(shù).(14分)
點(diǎn)評:本題主要考查了利用導(dǎo)數(shù)研究函數(shù)的單調(diào)性,以及利用導(dǎo)數(shù)求閉區(qū)間上函數(shù)的最值,屬于中檔題.