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$$\frac{2 \left(- \frac{6 x^{3}}{x^{3} - 1} + \frac{3 x \left(x^{2} - 1\right) \left(\frac{3 x^{3}}{x^{3} - 1} - 1\right)}{x^{3} - 1} + 1\right)}{x^{3} - 1} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -1 - \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1$$
$$\lim_{x \to 1^-}\left(\frac{2 \left(- \frac{6 x^{3}}{x^{3} - 1} + \frac{3 x \left(x^{2} - 1\right) \left(\frac{3 x^{3}}{x^{3} - 1} - 1\right)}{x^{3} - 1} + 1\right)}{x^{3} - 1}\right) = 0.222222222222222$$
$$\lim_{x \to 1^+}\left(\frac{2 \left(- \frac{6 x^{3}}{x^{3} - 1} + \frac{3 x \left(x^{2} - 1\right) \left(\frac{3 x^{3}}{x^{3} - 1} - 1\right)}{x^{3} - 1} + 1\right)}{x^{3} - 1}\right) = 0.222222222222222$$
- limits are equal, then skip the corresponding point
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
$$\left[- 2 \cos{\left(\frac{\pi}{9} \right)} - 1, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - 2 \cos{\left(\frac{\pi}{9} \right)} - 1\right]$$