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$$\frac{2 \left(\frac{2 x^{4} \left(\frac{8 x^{4}}{x^{4} + 1} - 3\right)}{x^{4} + 1} - \frac{8 x^{4}}{x^{4} + 1} + 1\right)}{x^{4} + 1} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -1.40662683528827$$
$$x_{2} = -0.5401828449376$$
$$x_{3} = 0.5401828449376$$
$$x_{4} = 1.40662683528827$$
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
$$\left[1.40662683528827, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -0.5401828449376\right] \cup \left[0.5401828449376, 1.40662683528827\right]$$