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Determine the function is concave upward and where it is concave downward
f(x)= x / (x+1)

  • Calculus -

    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

  • Calculus oops -

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

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