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

f(x)= x / (x+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

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

To determine if the function f(x) = x / (x+1) is concave upward or concave downward, we need to find the second derivative of the function and analyze its sign.

Step 1: Find the first derivative of f(x):
f'(x) = [(x+1)(1) - x(1)] / (x + 1)^2
= 1 / (x + 1)^2

Step 2: Find the second derivative of f(x):
f''(x) = [(x + 1)^2(0) - (1)(2 (x + 1)(1))] / (x + 1)^4
= -2 / (x + 1)^3

Step 3: Analyze the sign of the second derivative:
For concave upward, the second derivative should be positive (+).
For concave downward, the second derivative should be negative (-).

Since f''(x) = -2 / (x + 1)^3, we can conclude that for all x ≠ -1, the second derivative is negative. This means that the function f(x) = x / (x+1) is concave downward for all x ≠ -1.

Therefore, the function is concave downward everywhere except at x = -1 where it is undefined.