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$$\left(x + 1\right)^{x} \left(\frac{x}{x + 1} + \log{\left(x + 1 \right)}\right)^{2} + e^{x} - \frac{\left(x + 1\right)^{x} \left(\frac{x}{x + 1} - 2\right)}{x + 1} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -72.8720030830002$$
$$x_{2} = -118.872003083$$
$$x_{3} = -94.8720030830002$$
$$x_{4} = -44.8720030830002$$
$$x_{5} = -60.8720030830002$$
$$x_{6} = -106.872003083$$
$$x_{7} = -104.872003083$$
$$x_{8} = -80.8720030830002$$
$$x_{9} = -92.8720030830002$$
$$x_{10} = -56.8720030830002$$
$$x_{11} = -68.8720030830002$$
$$x_{12} = -90.8720030830002$$
$$x_{13} = -50.8720030830002$$
$$x_{14} = -62.8720030830002$$
$$x_{15} = -66.8720030830002$$
$$x_{16} = -114.872003083$$
$$x_{17} = -78.8720030830002$$
$$x_{18} = -70.8720030830002$$
$$x_{19} = -86.8720030830002$$
$$x_{20} = -84.8720030830002$$
$$x_{21} = -46.8720030830002$$
$$x_{22} = -74.8720030830002$$
$$x_{23} = -102.872003083$$
$$x_{24} = -120.872003083$$
$$x_{25} = -48.8720030830002$$
$$x_{26} = -64.8720030830002$$
$$x_{27} = -108.872003083$$
$$x_{28} = -98.8720030830002$$
$$x_{29} = -112.872003083$$
$$x_{30} = -76.8720030830002$$
$$x_{31} = -52.8720030830002$$
$$x_{32} = -100.872003083$$
$$x_{33} = -88.8720030830002$$
$$x_{34} = -110.872003083$$
$$x_{35} = -116.872003083$$
$$x_{36} = -96.8720030830002$$
$$x_{37} = -58.8720030830002$$
$$x_{38} = -54.8720030830002$$
$$x_{39} = -82.8720030830002$$
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:
Have no bends at the whole real axis