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

∵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+

1
x
,∴ea+
1
a
=0

g′(x)=-e-x+
1
x
,∴-e-b+
1
b
=0,
h′(x)=-e-x-
1
x
,∴-e-c-
1
c
=0,
∴a<0,b>0,c<0,
∵ea=-
1
a
,e-c=-
1
c
,相除得到0<
c
a
=e(a+c)<e0=1,∴a<c
综上a<c<b.
故选:A.