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$$\frac{4 x^{2} \left(\frac{8 x^{4}}{x^{4} + 1} - 3\right)}{\left(x^{4} + 1\right)^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 0$$
$$x_{2} = - \frac{\sqrt[4]{3} \cdot 5^{\frac{3}{4}}}{5}$$
$$x_{3} = \frac{\sqrt[4]{3} \cdot 5^{\frac{3}{4}}}{5}$$
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
$$\left(-\infty, - \frac{\sqrt[4]{3} \cdot 5^{\frac{3}{4}}}{5}\right] \cup \left[\frac{\sqrt[4]{3} \cdot 5^{\frac{3}{4}}}{5}, \infty\right)$$
Convex at the intervals
$$\left[- \frac{\sqrt[4]{3} \cdot 5^{\frac{3}{4}}}{5}, \frac{\sqrt[4]{3} \cdot 5^{\frac{3}{4}}}{5}\right]$$