f(x)=lnx-1/x-1的零点个数f(x)=lnx-(1/x-1)的零点个数
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![f(x)=lnx-1/x-1的零点个数f(x)=lnx-(1/x-1)的零点个数](/uploads/image/z/2535439-31-9.jpg?t=f%EF%BC%88x%EF%BC%89%3Dlnx-1%2Fx-1%E7%9A%84%E9%9B%B6%E7%82%B9%E4%B8%AA%E6%95%B0f%EF%BC%88x%EF%BC%89%3Dlnx-%EF%BC%881%2Fx-1%EF%BC%89%E7%9A%84%E9%9B%B6%E7%82%B9%E4%B8%AA%E6%95%B0)
f(x)=lnx-1/x-1的零点个数f(x)=lnx-(1/x-1)的零点个数
f(x)=lnx-1/x-1的零点个数
f(x)=lnx-(1/x-1)的零点个数
f(x)=lnx-1/x-1的零点个数f(x)=lnx-(1/x-1)的零点个数
f'(x)=1/x+1/(x-1)^2=[(x-1)^2+x]/(x-1)^2=(x^2-x+1)/(x-1)^2,(x>0,x≠1)
注意到x^2-x+1=(x-1/2)^2+3/4>0,则f'(x)>0,那么f(x)在定义域内单调递增,又(x->0+)f(x)=-∞,(x->1-)f(x)=+∞,(x->1+)f(x)=-∞,(x->∞)f(x)=+∞,
由零点定理知f(x)=lnx-1/(x-1)有2个零点,分别在(0,1),(1,+∞)内