[鎖定考向]
用導(dǎo)數(shù)解決函數(shù)的零點問題是近幾年高考命題的熱點題型之一.
常見的命題角度有:
(1)求函數(shù)零點或零點個數(shù);
(2)已知函數(shù)零點個數(shù)求參數(shù)的值或范圍.
[題點全練]
角度一:求函數(shù)零點或零點個數(shù)
1.已知函數(shù)f(x)=ax+ln x+1,討論函數(shù)f(x)零點的個數(shù).
解:法一:函數(shù)f(x)的定義域為(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(0,-1),斜率k=-a的直線.
當直線y=kx-1與函數(shù)u(x)=ln x的圖象相切時,兩者只有一個交點,此時設(shè)切點為P(x0,y0),
則解得
所以當k>1時,函數(shù)f(x)沒有零點;當k=1或k≤0時,函數(shù)f(x)有1個零點;當0<k<1時,函數(shù)f(x)有2個零點.
即當a<-1時,函數(shù)f(x)沒有零點;當a=-1或a≥0時,函數(shù)f(x)有1個零點;當-1<a<0時,函數(shù)f(x)有2個零點.
法二:函數(shù)f(x)的定義域為(0,+∞),
由f(x)=ax+ln x+1=0,得a=-.