函数f(x)=ex+lnx,g(x)=e-x+lnx,g(x)=e-x-lnx的零点分别是a,b,c,则( ) A.a<c<b B.c<b<a C.c<a<b D.b<a<c
问题描述:
函数f(x)=ex+lnx,g(x)=e-x+lnx,g(x)=e-x-lnx的零点分别是a,b,c,则( )
A. a<c<b
B. c<b<a
C. c<a<b
D. b<a<c
答
∵f(x)=ex+lnx,g(x)=e-x+lnx,g(x)=e-x-lnx的零点分别是a,b,c,
∴f′(x)=ex+
,∴ea+1 x
=0,1 a
g′(x)=-e-x+
,∴-e-b+1 x
=0,1 b
h′(x)=-e-x-
,∴-e-c-1 x
=0,1 c
∴a<0,b>0,c<0,
∵ea=-
,e-c=-1 a
,相除得到0<1 c
=e(a+c)<e0=1,∴a<cc a
综上a<c<b.
故选:A.