Calculus of a Single Variable, Chapter 3, 3.4, Section 3.4, Problem 7

Hello!
To determine where a function is concave upward or downward, we may use the second derivative. This function is defined everywhere, differentiable and second differentiable everywhere, so Concavity Theorem is applicable.
f'(x) = -24*(2x)/(x^2+12)^2 = -48*(x)/(x^2+12)^2,
f''(x) = -48*((x^2+12)^2 - x*[(x^2+12)^2]')/(x^2+12)^4 =
-48*((x^2+12)^2 - x*2*(x^2+12)*2x)/(x^2+12)^4 =
-48*(x^2+12 - 4x^2)/(x^2+12)^3 =-48*3*(4 - x^2)/(x^2+12)^3 =
= 48*3*(x^2-4)/(x^2+12)^3.
We see that f''(x)=0 for x=-2 and x=2. Moreover, it is negative inside (-2, 2) and is positive outside (the denominator is always positive).
The answer: f is concave upward at (-oo, -2) uuu(2, +oo) and is concave downward at (-2, 2).