Determine the function is concave upward and where it is concave downward

Question

f(x)= x / (x+1)

Answer 1

f ' (x) = ( x+1 - x)/(x+1)^2 = (x+1)^-2
f ''(x) = 2(x+1)-3
= 1/(x+1)^3

if x > -1 , f ''(x) is positive, so the curve is convace upwards
if x < -1, f ''(x) is negative, so the curve is concave downwards.

notice x ≠ -1, so there is a vertical asymptote at x + 1 = 0

Answer 2

I think you dropped a minus sign on f'', so the intervals are reversed.