考點(diǎn):利用導(dǎo)數(shù)研究函數(shù)的極值
專題:計(jì)算題,導(dǎo)數(shù)的綜合應(yīng)用
分析:A.求出導(dǎo)數(shù),求出y′=0,則x=±4,檢驗(yàn)在x=±4處附近導(dǎo)數(shù)符號,即可判斷;
B.求出導(dǎo)數(shù),由判別式小于0,即可判斷;
C.求出導(dǎo)數(shù),由于y′=3x2≥0,即可判斷;
D.求出導(dǎo)數(shù),y′=0,得x=0,檢驗(yàn)在x=0處附近導(dǎo)數(shù)的符號,即可判斷.
解答:
解:A.函數(shù)y=48x-x3的導(dǎo)數(shù)y′=48-3x2,y′=0,則x=±4,在x=±4處附近導(dǎo)數(shù)符號異號,則均為極值點(diǎn),故A正確;
B.函數(shù)y=x3-x2+x的導(dǎo)數(shù)y′=3x2-2x+1,判別式△=4-12<0,y′>0,函數(shù)單調(diào)遞增,故無極值,故B錯(cuò);
C.y=x3的導(dǎo)數(shù)y′=3x2≥0,函數(shù)單調(diào)遞增,無極值,故C錯(cuò);
D.函數(shù)y=ex-x的導(dǎo)數(shù)y′=ex-1,y′=0,得x=0,在x=0處附近導(dǎo)數(shù)左負(fù)右正,故為極小值點(diǎn),故D錯(cuò).
故選A.
點(diǎn)評:本題考查導(dǎo)數(shù)的運(yùn)用:求函數(shù)的極值,注意判斷導(dǎo)數(shù)在某點(diǎn)處的符號是否異號,屬于基礎(chǔ)題.