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$$2 \cdot \left(\frac{10 x}{\left(x^{2} + 4\right)^{2}} - \frac{1}{x^{2}}\right) = 0$$
Solve this equationThe roots of this equation
$$x_{1} = \frac{5}{2} + \sqrt{\frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} + \frac{59}{12}} + \sqrt{- 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} - \frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + \frac{85}{4 \sqrt{\frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} + \frac{59}{12}}} + \frac{59}{6}}$$
$$x_{2} = - \sqrt{- 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} - \frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + \frac{85}{4 \sqrt{\frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} + \frac{59}{12}}} + \frac{59}{6}} + \frac{5}{2} + \sqrt{\frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} + \frac{59}{12}}$$
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
$$\left[- \sqrt{- 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} - \frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + \frac{85}{4 \sqrt{\frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} + \frac{59}{12}}} + \frac{59}{6}} + \frac{5}{2} + \sqrt{\frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} + \frac{59}{12}}, \frac{5}{2} + \sqrt{\frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} + \frac{59}{12}} + \sqrt{- 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} - \frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + \frac{85}{4 \sqrt{\frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} + \frac{59}{12}}} + \frac{59}{6}}\right]$$
Convex at the intervals
$$\left(-\infty, - \sqrt{- 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} - \frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + \frac{85}{4 \sqrt{\frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} + \frac{59}{12}}} + \frac{59}{6}} + \frac{5}{2} + \sqrt{\frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} + \frac{59}{12}}\right] \cup \left[\frac{5}{2} + \sqrt{\frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} + \frac{59}{12}} + \sqrt{- 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} - \frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + \frac{85}{4 \sqrt{\frac{8}{9 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}}} + 2 \sqrt[3]{\frac{5 \sqrt{1257}}{144} + \frac{547}{432}} + \frac{59}{12}}} + \frac{59}{6}}, \infty\right)$$