高中数学哪个省份用北师大版-哪个app学高中数学好
21.(12分)已知函数
f(x)?
1
?x?alnx
.
x
(1)讨论
f(x)
的单调性;
(2)若
f(x)存在两个极值点
x
1
,x
2
,证明:
f
?x
1
?
?f
?
x
2
?
?a?2
.
x
1
?x
2
1?1a?x
2
?ax?1,x?0
证明:
f(x)??x?alnx?f
?
(x)?
2
?1??
xxxx
2
?
??a
2
?4?0
?
?
?
a
?a?0,x
1
?x
2
?a,x
1
x
2
?1
?
?0
?
2
设0?x
1
?x
2
,则0?x
1
?1?x
2
欲证<
br>f(x
1
)?f(x
2
)
?a?2?f(x
1
)?f(x
2
)?(a?2)(x
1
?x
2
)
x
1
?x
2
?f(x
1
)?f(x
2)?(2?a)(x
1
?x
2
)?0
?(
?
1
1
?)?(x
1
?x
2
)?a(lnx
1
?lnx
2
)?(2?a)(x
1
?x
2
)?0
x
1
x
2
x
2
?x
1
?(x
2
?x
1
)?a(lnx
1
?lnx
2
)?(a?2)(x
2
?x
1
)?0
x
1
x
2
a
(
lnx
1
?lnx
2
)?(a?2)?0
x
2
?x
1
?1?1?
?1?1?
?
a
(lnx
1
?lnx
2
)?(a?2)?0
x
2
?x
1
1(lnx
1
?lnx
2
)?1?0?lnx
1
?lnx
2
?x
2
?x
1
?0
x
2
?x<
br>1
?2lnx
1
?(lnx
1
?lnx
2
)?
?2lnx
1
?
x
2
x
1
1
?x
1
?0?2lnx
1
?lnx
1
x
2
??x
1
?0
x
1
x
1
1
?x<
br>1
?0,0?x
1
?1
x
1
1
?
x?0,0?x?1
x
21?x
2
?2x?1?(x?1)
2
?
g
?
(x)??
2
?1???0
xxx
2x
2
?y?g(x)在(0,1)上单调递减??g(x)?g(0)?0
下面证
明g(x)?2lnx?
?2lnx
1
?
1
?x
1
?0在(0,1)上成立,即原命题成立(证毕)
x
1