f(x)=log^2(x+1)/(x-1)+log^2(x-1)+log^2(p-x)的值域(负无穷,log^(p+1)
1个回答

f(x)=log^2(x+1)/(x-1)+log^2(x-1)+log^2(p-x)

定义域:(x+1)/(x-1)>0,且x-1>0,且p-x>0

x<-1或x>1,且x>1,且x<p

∴1<x<p

f(x)=log2 (x+1)/(x-1)+log2 (x-1)+log2 (p-x)

=log2 [(x+1)/(x-1) * (x-1) * (p-x) ]

=log2 [(x+1) (p-x) ]

=log2[-x^2+(p-1)x+p]

令g(x)=-x^2+(p-1)x+p,其最大值:[4*(-1)*p-(p-1)^2]/[4*(-1)]=(p+1)^2/4恒大于0

另外,对称轴x=(p-1)/2应符合定义域1<x<p的要求:

1<(p-1)/2<p

2<p<2p+1

故P>3