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$$\frac{\frac{2 \log{\left(x \right)}}{\left(x + 2\right)^{2}} - \frac{2}{x \left(x + 2\right)} - \frac{1}{x^{2}}}{x + 2} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 24294.3612637142$$
$$x_{2} = 55296.0175240638$$
$$x_{3} = 37615.0545701149$$
$$x_{4} = 29835.2864925341$$
$$x_{5} = 36504.5395380226$$
$$x_{6} = 42051.3926649508$$
$$x_{7} = 48686.0781758374$$
$$x_{8} = 26505.142019564$$
$$x_{9} = 45371.8340832042$$
$$x_{10} = 54196.0466546917$$
$$x_{11} = 59689.4020575102$$
$$x_{12} = 43158.8847440741$$
$$x_{13} = 44265.7017725604$$
$$x_{14} = 51994.1010647193$$
$$x_{15} = 35393.5551813782$$
$$x_{16} = 57493.9931924793$$
$$x_{17} = 46477.2757599979$$
$$x_{18} = 34282.1802372611$$
$$x_{19} = 33170.5159341437$$
$$x_{20} = 40943.2398361897$$
$$x_{21} = 53095.4101884672$$
$$x_{22} = 27613.9915244101$$
$$x_{23} = 60786.1652576388$$
$$x_{24} = 30946.8770750174$$
$$x_{25} = 25398.3042276186$$
$$x_{26} = 28724.2019017552$$
$$x_{27} = 39834.446294619$$
$$x_{28} = 58592.0142527443$$
$$x_{29} = 56395.3303766332$$
$$x_{30} = 32058.6923765109$$
$$x_{31} = 47582.0239274937$$
$$x_{32} = 38725.0391970573$$
$$x_{33} = 7.75850139245233$$
$$x_{34} = 49789.4400935317$$
$$x_{35} = 50892.1128862442$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -2$$
$$\lim_{x \to -2^-}\left(\frac{\frac{2 \log{\left(x \right)}}{\left(x + 2\right)^{2}} - \frac{2}{x \left(x + 2\right)} - \frac{1}{x^{2}}}{x + 2}\right) = - \infty \operatorname{sign}{\left(1.38629436111989 + 2 i \pi \right)}$$
$$\lim_{x \to -2^+}\left(\frac{\frac{2 \log{\left(x \right)}}{\left(x + 2\right)^{2}} - \frac{2}{x \left(x + 2\right)} - \frac{1}{x^{2}}}{x + 2}\right) = \infty \operatorname{sign}{\left(1.38629436111989 + 2 i \pi \right)}$$
- the limits are not equal, so
$$x_{1} = -2$$
- is an inflection 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
$$\left[7.75850139245233, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 7.75850139245233\right]$$