内容正文:
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f(x)
!
[0,π]"#1$%&,
'
f(x)
()*+
,
,-
f(x)
!
[-π,π]"#2$%&,
.",/
,
0
0<a≤
2
π2
1
,f(x)
!
[-π,π]"#
2
$%&
,
0
a> 2
π2
1
,f(x)
!
[-π,π]"2%&.
21.(1)
!
:
345
,
*+
f(x)
6789(
(0,
+∞),:;*+ f′(x)=lnx-a(x-1).
<
h(x)=f′(x),
=
h′(x)=1-ax
x
,
0
a≤01,h′(x)=
1-ax
x
>0
>?@
,
,-
h(x)
!
(0,+∞)"ABCD,E h(1)=0,
,-FGH5
x∈(1,+∞),I# h(x)=f′(x)
>0,
J45KL
,
M
a≤0K?@.
0
a>0
1
,
N
x (∈ 0,1)a ,= h′(x)>0;
N
x (∈ 1a,+ )∞ ,= h′(x)<0.
,-
h(x) (! 0,1)a " A B C D, (! 1a,
+ )∞ "ABCO.
,-
h(x)max (=h 1)a =-lna+a-1=0.
P
g(a)=-lna+a-1,
=
g′(a)=a-1
a
.
0
0<a<1
1
,g′(a)<0;
0
a>1
1
,g′(a)>
0,
,-
g(a)
!
(0,1)
"ABCO
,
!
(1,+∞)"AB
CD
.
,-
g(a)≥g(1)=0,M a=1.
(2)
"#
:
0
a=1
1
,f(x)=xlnx-1
2
x2,
=
f′(x)=1+lnx-x.
3
(1)
Q
f′(x)≤0>?@,
,-
f(x)
!
(0,+∞)"ABCO,
E
f(1)=-1
2
,f(x1)+f(x2)=-1=2f(1).
KR<
0<x1 <x2,= f(x1)>f(x2),
,-
f(x1)>f(1),f(x2)<f(1),
,-
0<x1 <1<x2,
ST
x1+x2 >2,UVT x2 >2-x1.
W(
f(x)
!
(0,+∞)"ABCO,
,-UVT
f(x2)<f(2-x1),
'
f(x1)+f(x2)=-1,
,-UVT
-1-f(x1)<f(2-x1),
X
f(2-x1)+f(x1)>-1.
P
F(x)=f(x)+f(2-x)(
!"
x∈(0,1)),
=
F(1)=-1.
,-ST
f(2-x1)+f(x1)>-1,
UVT
F(x)>F(1),x∈(0,1),
F′(x)=f′(x)-f′(2-x)
=1+lnx-x-[1+ln(2-x)-2+x]
=lnx-ln(2-x)+2(1-x),x∈(0,1),
P
m(x)=F′(x),
=
m′(x)=2(1-x)
2
x(2-x)
>0,x∈(0,1),
,-
F′(x)
!YZ
(0,1)
"ABCD
,
,-
F′(x)<F′(1)=0,
,-*+
F(x)=f(x)+f(2-x)
!YZ
(0,1)
"
ABCO
,
,-
F(x)>F(1),x∈(0,1),M x1+x2 >2.
22.(1)
!
:f(x)=xlnx-aex
6789(
(0,
+∞),f′(x)=lnx+1-aex,
=
f(x)
!
(0,+∞)"[!\$]^&_`G
f′(x)=0
!
(0,+∞)"#\$K_ab,
3
f′(x)=lnx+1-aex =0,
cd
a=lnx+1
ex
,
P
g(x)=lnx+1
ex
,
=
g′(x)=
1
x
-(lnx+1)
ex
,
P
h(x)= 1
x
-lnx-1,
=
h′(x)=-1
x2
-1
x
,
0
x>0
1
,h′(x)<0,
M*+
h(x)
!
(0,+∞)
"ABCO
,
E
h(1)=0,
,-0
x∈(0,1)1,h(x)>0,g′(x)>0,g(x)
ABCD
;
0
x∈(1,+∞)1,h(x)<0,g′(x)<0,g(x)A
BCO
,
,-
g(x)
!
x=1
efd]g^
,
hijg^
,
g(x)max =g(1)=
1
e
,
,-
a< 1
e
,
'0
x→01,g(x)→-∞;
0
x→+∞ 1,g(x)gG0EklG0,
,-
a (6f^mni 0,1)e .
(2)
"#
:φ(x)=lnx+1-f′(x)=lnx+1-
(lnx+1-aex)=aex,
3 φ(1)=e,d a=1,= φ(x)=ex,
oT φ(x0)<p(1)<y0?@,
UVT
e
x1+x2
2 <k=e
x2-ex1
x2-x1
<e
x1+ex2
2
,
X
e
x2+x1
2 ·e-x1 <e
x2-ex1
x2-x1
·e-x1 <e
x1+ex2
2
·e-x1,
X
e
x2-x1
2 <e
x2-x1