已知函數(shù)f(x)=x3+ax2+bx+c.f(x)在點x=0處取得極值,并且在單調(diào)區(qū)間[0,2]和[4,5]上具有相反的單調(diào)性.
(1)求實數(shù)b的值;
(2)求實數(shù)a的取值范圍.
【答案】
分析:(1)根據(jù)f(x)在點x=0處取得極值,可得f'(0)=0,建立等量關(guān)系,求出參數(shù)b即可.
(2)有條件“在單調(diào)區(qū)間[0,2]和[4,5]上具有相反的單調(diào)性”可知函數(shù)的極值點應(yīng)介于[2,4]即可.
解答:解:(1)f'(x)=3x
2+2ax+b,因為f(x)在點x=0處取得極值,
所以f'(x)=0,即得b=0;
(2)令f'(0)=0,即3x
2+2ax=0,
解得x=0或
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214328944839749/SYS201310232143289448397017_DA/0.png)
.
依題意有
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214328944839749/SYS201310232143289448397017_DA/1.png)
.
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214328944839749/SYS201310232143289448397017_DA/images2.png)
因為在函數(shù)在單調(diào)區(qū)間[0,2]和[4,5]上具有相反的單調(diào)性,所以應(yīng)有
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214328944839749/SYS201310232143289448397017_DA/2.png)
,
解得-6≤a≤-3.
點評:本小題主要考查運用導(dǎo)數(shù)研究函數(shù)的單調(diào)性及極值等基礎(chǔ)知識,考查綜合分析和解決問題的能力.