设t＝(x²-x+1)/(x²+x+1)
→(t-1)x²+(t+1)x+t-1＝0.
△＝(t+1)²-4(t-1)²≥0
→(3t-1)(t-3)≤0
→1/3≤t≤3.
故所求最大值为3,所求最小值为1/3.