fx=3x/x2+x+1(x>0)的值域

Question

fx=3x/x2+x+1(x>0)的值域

Answer 1

f(x)=3x/(x^2+x+1)=3x/[(x-1/2)^2+3/4]
x＞0时,f(x)＞0
令f(x)=3x/(x^2+x+1)=m
mx^2+(m-3)x+m=0
判别式=(m-3)^2-4m*m≥0
3m^2+6m-9≤0
(m+3)(m-1)≤0
-3≤m≤1,即：
-3≤f(x)≤1
又：f(x)＞0,所以：
0＜f(x)≤1
值域（0,1]