Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative$$\frac{2 \left(x - 1\right)}{\left(\left(x - 1\right)^{2} + 1\right)^{2}} - \frac{1}{x^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = \frac{\sqrt{5}}{2} + \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} + \frac{3}{2}$$
$$x_{2} = - \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} + \frac{\sqrt{5}}{2} + \frac{3}{2}$$
Сonvexity and concavity intervals:Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[- \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} + \frac{\sqrt{5}}{2} + \frac{3}{2}, \frac{\sqrt{5}}{2} + \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} + \frac{3}{2}\right]$$
Convex at the intervals
$$\left(-\infty, - \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} + \frac{\sqrt{5}}{2} + \frac{3}{2}\right] \cup \left[\frac{\sqrt{5}}{2} + \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} + \frac{3}{2}, \infty\right)$$