考點(diǎn):函數(shù)的零點(diǎn)
專題:函數(shù)的性質(zhì)及應(yīng)用
分析:先求出f(x)的最小值,通過討論a的范圍,結(jié)合二次函數(shù)的性質(zhì),從而得到函數(shù)零點(diǎn)的個數(shù).
解答:
解:∵f(x)=(x+1)
2+a-2,對稱軸x=-1,
∴f(x)在(-∞,-1)遞減,在(-1,
]遞增,
∴f(x)
min=a-2,
當(dāng)a-2>0,即a>2時,f(x)無零點(diǎn),
當(dāng)a-2=0,即a=2時,f(x)與x軸有1個交點(diǎn)(-1,0),
當(dāng)a-2<0,即a<2時,令f(x)=0,解得x=-1±
,
令-1+
>
,解得:a<-
,
∴a<-
時,f(x)與x軸有1個交點(diǎn)(-1-
,0),
-
≤a<2時,f(x)與x軸有2個交點(diǎn)(-1-
,0),(-1+
,0),
綜上,a>0時,函數(shù)f(x)無零點(diǎn),
a=2時,函數(shù)f(x)有一個零點(diǎn)x=-1,
-
≤a<2時,函數(shù)f(x)有2個零點(diǎn)x=-1±
,
a<-
時,函數(shù)f(x)有1個零點(diǎn)x=-1-
.
點(diǎn)評:本題考查了函數(shù)的零點(diǎn)問題,考查了二次函數(shù)的性質(zhì),考查了分類討論思想,是一道中檔題.