#f(x) = x/(x-2)#

The domain is #D = RR-{2}#

#f'(x) = ((x)'(x-2) - x(x-2)')/(x-2)^2#

#f'(x) = (x-2-x)/(x-2)^2 = -2/(x-2)^2#

To find the local max an min of #f(x)#, you need to study the sign of the derivative :

You are absolutely right, there aren't any local max or min.

#f''(x) = ((-2)'(x-2)^2-(-2)((x-2)^2)')/(x-2)^4#

#f''(x) = (4(x-2))/(x-2)^4 = 4/(x-2)^3#

To find the intervals in which #f(x#) is concave up or concave down, you need to study the sign of the second derivative :

The function is concave down for #x in ]-oo;+2[# and is concave up for #x in ]+2;+oo[#.