[鎖定考向]
用導(dǎo)數(shù)解決函數(shù)的零點(diǎn)問題是近幾年高考命題的熱點(diǎn)題型之一.
常見的命題角度有:
(1)求函數(shù)零點(diǎn)或零點(diǎn)個(gè)數(shù);
(2)已知函數(shù)零點(diǎn)個(gè)數(shù)求參數(shù)的值或范圍.
[題點(diǎn)全練]
角度一:求函數(shù)零點(diǎn)或零點(diǎn)個(gè)數(shù)
1.已知函數(shù)f(x)=ax+ln x+1,討論函數(shù)f(x)零點(diǎn)的個(gè)數(shù).
解:法一:函數(shù)f(x)的定義域?yàn)?/span>(0,+∞),由f(x)=ax+ln x+1=0,得ln x=-ax-1,
令u(x)=ln x,v(x)=-ax-1,則函數(shù)v(x)的圖象是過定點(diǎn)(0,-1),斜率k=-a的直線.
當(dāng)直線y=kx-1與函數(shù)u(x)=ln x的圖象相切時(shí),兩者只有一個(gè)交點(diǎn),此時(shí)設(shè)切點(diǎn)為P(x0,y0),
則解得
所以當(dāng)k>1時(shí),函數(shù)f(x)沒有零點(diǎn);當(dāng)k=1或k≤0時(shí),函數(shù)f(x)有1個(gè)零點(diǎn);當(dāng)0<k<1時(shí),函數(shù)f(x)有2個(gè)零點(diǎn).
即當(dāng)a<-1時(shí),函數(shù)f(x)沒有零點(diǎn);當(dāng)a=-1或a≥0時(shí),函數(shù)f(x)有1個(gè)零點(diǎn);當(dāng)-1<a<0時(shí),函數(shù)f(x)有2個(gè)零點(diǎn).
法二:函數(shù)f(x)的定義域?yàn)?/span>(0,+∞),
由f(x)=ax+ln x+1=0,得a=-.