g(x)=2a^2lnx+1/2*ax^2-x
g(x)'=(a*x^2-x+2a^2)/x 有两个极值
则x1+x2=1/a x1*x2=2a a*x^2=x2-2*a^2
1- 8a^3<0 a<1/2,又x1>0,x2>0 则a>0,则 0<a<1/2
不妨令x2>x1 则x2/x1>1
p(x) =lnx-x+1 p(x)'=1/x-1 当x>1时单调递减
则p(x)>p(1)=0 在x>1时恒成立
则ln(x2/x1)<x2/x1-1 =(x2-x1)/x1
则 k=(g(x2)-g(x1))/(x2-x1)
=2a^2(lnx2-lnx1)/(x2-x1)+1/2*a*(x1+x2)-1
<2a^2(x2-x1)/x1/(x2-x1)-1/2
=2a^2/x1-1/2<1*(1/2)^2/x1-1/2=1/2(x1）-1/2<1/x1+1/x2
则1/x1+1/x2>k
g(x)'=(a*x^2-x+2a^2)/x 有两个极值
则x1+x2=1/a x1*x2=2a a*x^2=x2-2*a^2
1- 8a^3<0 a<1/2,又x1>0,x2>0 则a>0,则 0<a<1/2
不妨令x2>x1 则x2/x1>1
p(x) =lnx-x+1 p(x)'=1/x-1 当x>1时单调递减
则p(x)>p(1)=0 在x>1时恒成立
则ln(x2/x1)<x2/x1-1 =(x2-x1)/x1
则 k=(g(x2)-g(x1))/(x2-x1)
=2a^2(lnx2-lnx1)/(x2-x1)+1/2*a*(x1+x2)-1
<2a^2(x2-x1)/x1/(x2-x1)-1/2
=2a^2/x1-1/2<1*(1/2)^2/x1-1/2=1/2(x1）-1/2<1/x1+1/x2
则1/x1+1/x2>k
