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$$\frac{- \frac{20 x^{2} \left(10 - \frac{10 x^{2} + 9}{x^{2} + 1}\right)}{10 x^{2} + 9} + \frac{2 x^{2} \left(10 - \frac{10 x^{2} + 9}{x^{2} + 1}\right)}{x^{2} + 1} - \frac{40 x^{2}}{x^{2} + 1} + \frac{4 x^{2} \left(10 x^{2} + 9\right)}{\left(x^{2} + 1\right)^{2}} + 10 - \frac{10 x^{2} + 9}{x^{2} + 1}}{10 x^{2} + 9} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = - \sqrt{- \frac{19}{60} + \frac{\sqrt{1441}}{60}}$$
$$x_{2} = \sqrt{- \frac{19}{60} + \frac{\sqrt{1441}}{60}}$$
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
$$\left[- \sqrt{- \frac{19}{60} + \frac{\sqrt{1441}}{60}}, \sqrt{- \frac{19}{60} + \frac{\sqrt{1441}}{60}}\right]$$
Convex at the intervals
$$\left(-\infty, - \sqrt{- \frac{19}{60} + \frac{\sqrt{1441}}{60}}\right] \cup \left[\sqrt{- \frac{19}{60} + \frac{\sqrt{1441}}{60}}, \infty\right)$$