We must to find all values of m which df(x) >=0, for every x belongs to inteval (1, +infty).
First cases. The function f(x) is increasing for x < -m/3) and for x>-m/3 if and only if the numerator of df is nonnegative.
Secend cases. When discriminant := discrim(numer(df), x) >0, Suppose x_1, x_2 are different solutions of numer(df), x_1<x_2. We must find m so that x_2 >=1 and -m/3<=1.
From two cases, I think, our anserw are -3<=m<=1 or m >=3/2.
This is my code (i don't sure it is right).
restart;f:=x->(x^2 + 2*m*x + m+2)/(3*x+m):
This is my explain. Please comment to me. Thank you.