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$\displaystyle f'(x)=\frac{6x}{(x^2+3)^2}$

$\displaystyle f"(x)=\frac{6(x^2+3)-24x^2(x^2+3)}{(x^2+3)^2}$

$\displaystyle f"(x)=\frac{6-24x^2}{(x^2+3)}$

x= + or - 1/2 and + or - sqrt(3). I was going to do the sign chart/number line method where you plug in points, by my intervals are different.