What are the points of inflection of f(x)=1/(1+x^2)?

The inflection points could be found by calculating the roots of the second derivative (if there are any).

Hence we have that

${d}^{2} f \frac{x}{\mathrm{dx}} ^ 2 = 0 \implies \frac{6 {x}^{2} - 2}{{x}^{2} + 1} ^ 3 = 0 \implies x = \frac{1}{\sqrt{3}} \mathmr{and} x = - \frac{1}{\sqrt{3}}$

So the inflection points are

$\left(\frac{1}{\sqrt{3}} , f \left(\frac{1}{\sqrt{3}}\right)\right)$ and $\left(- \frac{1}{\sqrt{3}} , f \left(- \frac{1}{\sqrt{3}}\right)\right)$