解:(1)a=1时, , 则![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073855-64653.png) 令F"(x)≥0有:x≤0(舍去)或x≥1; 令F"(x)≤0有0≤x≤1 故F(x)的单增区间为[1,+∞);单减区间为(0,1]. (2)构造F(x)=f(x)﹣g(x)(x>1), 即![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073855-55599.png) 则 . ①当a≤e时,ex﹣a>0成立,则x>1时,F"(x)>0,即F(x)在(1,+∞)上单增, 令F(1)=e﹣a﹣a≥0,∴ ,故![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019073856-27195.png) ②a>e时,F"(x)=0有x=1或x=lna>1 令F"(x)≥0有x≤1或x≥lna; 令F"(x)≤0有1≤x≤lna 即F(x)在(1,lna]上单减;在[lna,+∞)上单增 故F(x)min=F(lna)=﹣aln(lna)﹣a>0, ∴ ,舍去 综上所述,实数a的取值范围 |