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$$- \frac{\frac{\left(1 + \frac{3}{x^{2}}\right)^{2}}{x + 3 - \frac{3}{x}} + \frac{6}{x^{3}}}{x + 3 - \frac{3}{x}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = - \frac{\sqrt{-8 + \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}}}{2} + \frac{\sqrt{-16 - 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}} - \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + \frac{36}{\sqrt{-8 + \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}}}}}{2}$$
$$x_{2} = - \frac{\sqrt{-16 - 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}} - \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + \frac{36}{\sqrt{-8 + \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}}}}}{2} - \frac{\sqrt{-8 + \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}}}{2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(- \frac{\frac{\left(1 + \frac{3}{x^{2}}\right)^{2}}{x + 3 - \frac{3}{x}} + \frac{6}{x^{3}}}{x + 3 - \frac{3}{x}}\right) = \infty$$
$$\lim_{x \to 0^+}\left(- \frac{\frac{\left(1 + \frac{3}{x^{2}}\right)^{2}}{x + 3 - \frac{3}{x}} + \frac{6}{x^{3}}}{x + 3 - \frac{3}{x}}\right) = \infty$$
- 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
$$\left[- \frac{\sqrt{-16 - 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}} - \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + \frac{36}{\sqrt{-8 + \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}}}}}{2} - \frac{\sqrt{-8 + \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}}}{2}, - \frac{\sqrt{-8 + \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}}}{2} + \frac{\sqrt{-16 - 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}} - \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + \frac{36}{\sqrt{-8 + \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}}}}}{2}\right]$$
Convex at the intervals
$$\left(-\infty, - \frac{\sqrt{-16 - 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}} - \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + \frac{36}{\sqrt{-8 + \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}}}}}{2} - \frac{\sqrt{-8 + \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}}}{2}\right] \cup \left[- \frac{\sqrt{-8 + \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}}}{2} + \frac{\sqrt{-16 - 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}} - \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + \frac{36}{\sqrt{-8 + \frac{2}{\sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}} + 2 \sqrt[3]{\frac{3 \sqrt{3801}}{4} + \frac{185}{4}}}}}}{2}, \infty\right)$$