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$$\frac{2 \left(\frac{2 \log{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{2}{x \left(x + 1\right)} - \frac{1}{x^{2}}\right)}{x + 1} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 46775.8585929223$$
$$x_{2} = 58918.2997675932$$
$$x_{3} = 42334.6825768463$$
$$x_{4} = 52307.2498678697$$
$$x_{5} = 33405.1746958615$$
$$x_{6} = 40108.3596863583$$
$$x_{7} = 38993.7096045184$$
$$x_{8} = 57818.3400272034$$
$$x_{9} = 55616.2085554841$$
$$x_{10} = 50097.2474900641$$
$$x_{11} = 43446.3853380967$$
$$x_{12} = 41222.0125734309$$
$$x_{13} = 31162.6486362084$$
$$x_{14} = 6.25101515538997$$
$$x_{15} = 56717.6479698102$$
$$x_{16} = 32284.4059538666$$
$$x_{17} = 37878.049873297$$
$$x_{18} = 44557.1374212396$$
$$x_{19} = 34524.9318657011$$
$$x_{20} = 48990.9866669728$$
$$x_{21} = 51202.6628244769$$
$$x_{22} = 35643.6652135493$$
$$x_{23} = 36761.3705751831$$
$$x_{24} = 45666.9559962362$$
$$x_{25} = 47883.8629099866$$
$$x_{26} = 54514.006347851$$
$$x_{27} = 60017.5418322687$$
$$x_{28} = 53411.0255203816$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$\lim_{x \to -1^-}\left(\frac{2 \left(\frac{2 \log{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{2}{x \left(x + 1\right)} - \frac{1}{x^{2}}\right)}{x + 1}\right) = - \infty i$$
$$\lim_{x \to -1^+}\left(\frac{2 \left(\frac{2 \log{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{2}{x \left(x + 1\right)} - \frac{1}{x^{2}}\right)}{x + 1}\right) = \infty i$$
- the limits are not equal, so
$$x_{1} = -1$$
- 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
$$\left[6.25101515538997, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 6.25101515538997\right]$$