f(x)=ax-3x²/2,
最大值=a^2/6<=1/6,
a^2<=1,
-1<=a<=1.
f(1/4)=(8a-3)/32>=1/8,a>=7/8
f(1/2)=(3a-1)/6>=1/8.a>=7/12.
综上所述,实数a的范围为:7/8<=a<1