#晒收益#

Dear Peter,

I'm sorry to hear that you've been stuck on a math problem. When a = 1, the function f(x)=x+7x + 3lnx. First, we find its derivative f′(x)=2x + 7+3/x=(2x+7x + 3)/x=((2x + 1)(x+3))/x. Since x>0 (domain of lnx), f′(x)>0 alw，For the second - part, when 2≤x≤3e, f(x)=ax+(a + 6)x+3lnx≥ - 6 holds constantly. That is ax+(a + 6)x+3lnx + 6≥0. We can analyze it by cases of a. If a≥0, y = ax+(a + 6)x+3lnx+6 is increasing on [2,3e], and we can verify it meets the requirement. If a<0, we need to further study the extreme - value points of the function by its derivative and find the range of a that satisfies the condition. Hope these hints can help you solve the problem.

Look forward to your early reply.

Yours,