Sometimes replacing a variable with another expression can help to simplify the expression.
If I replace x in " f(x) for x>=1 " using x=u+1, I get f(u+1), u>=0.
When I then take d/dx f(x), and then set m=3 for example,
Looking at this output I can't really see that it's positive for x>=1.
But this is easy to see if I do x = u+1, where x>=1 means u>=0:
Here it is easy to see that for m=3, d/du f(u+1)>=0 for u>=0 and so d/dx f(x)>=0 for x>=1.
so the derivative of f with m=4 is positive for x> 5+2*sqrt(6).