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$$\frac{\frac{\left(1 + \frac{2}{\log{\left(x \right)} + 1}\right) \left(\log{\left(x \right)} - 1\right)}{\log{\left(x \right)} + 1} - 1 - \frac{2}{\log{\left(x \right)} + 1}}{x^{2} \left(\log{\left(x \right)} + 1\right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = e^{-3}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0.367879441171442$$
$$\lim_{x \to 0.367879441171442^-}\left(\frac{\frac{\left(1 + \frac{2}{\log{\left(x \right)} + 1}\right) \left(\log{\left(x \right)} - 1\right)}{\log{\left(x \right)} + 1} - 1 - \frac{2}{\log{\left(x \right)} + 1}}{x^{2} \left(\log{\left(x \right)} + 1\right)}\right) = \infty$$
$$\lim_{x \to 0.367879441171442^+}\left(\frac{\frac{\left(1 + \frac{2}{\log{\left(x \right)} + 1}\right) \left(\log{\left(x \right)} - 1\right)}{\log{\left(x \right)} + 1} - 1 - \frac{2}{\log{\left(x \right)} + 1}}{x^{2} \left(\log{\left(x \right)} + 1\right)}\right) = -\infty$$
- the limits are not equal, so
$$x_{1} = 0.367879441171442$$
- 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
$$\left[e^{-3}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, e^{-3}\right]$$