Question

已知定义在R上的偶函数f(x)在（0.+∞）上为减函数，且f(1/2)=0,

则满足f（log(1/2)x)<0的x的取值范围是？

答案是零到二分之根二并上根二到正无穷

Answer 1

当log(1/2)x>0时
f（log(1/2)x)<0
f（log(1/2)x)<f(1/2)
f(x)在（0.+∞）上为减函数
log(1/2)x>1/2
所以
0<x<(1/2)^1/2=√2/2
偶函数f(x)在（0.+∞）上为减函数
所以f(-x)在(-∞,0)上是增函数，f(1/2)=0，f(-1/2)=0
当log(1/2)x<0时
f（log(1/2)x)<0
f(log(1/2)x)<f(-1/2)
log(1/2)x<-1/2
x>1/2^(-1/2)=√2
所以综上
x>√2或0<x<√2/2

Answer 2

解：需满足log(1/2)x>1/2或log(1/2)x<-1/2
解得0<x<√2/2 或x>√2