(3)若f(x) xm(1 x)n,m与n为正整数
当x 0,1 时,f(x) 0
f (x) mxm 1(1 x)n xmn(1 x)n 1( 1)
n mm 1n 1
x(1 x)m(1 x)
mm
)时,f (x) 0 n m
m
)内单调递减 f(x)在(0,
n mm
,1)时,f (x) 0 当x (
n m
m
,1)内单调递减。
f(x)在(
n m
当x (0,
x (
m
,1)f (x) 0n m
f maxf(x)
0 x 1
m
max f(0),f()
n m
mm
nn
(m n)m n
f
f(x)dx1
010
1
xm(1 x)ndx
2(sin2t)m(1 sin2t)ndsin2t
2sin2mtcos2ntcost 2
sintdt
n!m!(n m 1)!
f
2
[ x(1 x)dx]
04m
4n
2
1
2m2n
12
12
[ sintcostd(sint)]
20
[ 22sin
4m 1
tcos
4n 1
tdt]
12