◎◎◎◎简单的数学问题◎◎◎◎

分子分母同除m
1/(m+1/m)
用均值不等式
当m>0
m+1/m>=2
当m<0
m+1/m<=-2
所以1/(m+1/m)∈【-1/2，1/2】
即m/(m^2+1)的取值范围∈【-1/2，1/2】

m>o时，m/(m^2+1）=1/(m+1/m)
m+1/m>=2,则0<1/(m+1/m)<=1/2
又这是一个奇函数，且m可以为0
故-1/2=<m<=1/2

解：y=m/(m^2+1)
又m^2+1≥2m,m^2+1≥-2m
-1/2≤y≤1/2

[-1/2,1/2]

设m/(m^2+1)=p
则：
pm^2-m+p=0
判别式△=1-4p^2≥0
p^2≤1/4
-1/2≤p≤1/2

m/(m^2+1)的取值范围：[-1/2,1/2]

m/(m^2+1)
=1/(m+1/m)
1.m不等于0
（1）m>0,
m+1/m>=2*根号（m*1/m)=2
故0<1/(m+1/m)<=1/2
(2) m<0
m+1/m=-[(-m)+(-1/m)]>=2*根号[(-m)*(-1/m)]=2,m+1/m<=-2
故-1/2<=m+1/m<0
2.m=0,m/(m^2+1)=0
综上所述 -1/2<=m/(m^2+1)<=1/2