已知函数f(x)=lnx, g(x)= 2 a 2 x 2 (a>0) (Ⅰ)若设F(x)=f(x)+g(x),求F(x)的单调递增区间; (Ⅱ)若函数 H(x)=f(x)+ 2g(x) 图象上任意点处的切线的斜率k≤1恒成立,求实数a的最小值; (Ⅲ)是否存在实数m,使得函数 p(x)= 1 3 x 3 + x 2 +m- 2 3 的图象与 q(x)= 3 2 f( x 2 ) 的图象恰好有三个不同的交点?若存在,求出m的取值范围,若不存在,说明理由.
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