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$$\frac{24 x - \frac{\left(12 x^{2} + e^{x}\right)^{2}}{4 x^{3} + e^{x}} + e^{x}}{4 x^{3} + e^{x}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 77.198731330601$$
$$x_{2} = 77.0124850969537$$
$$x_{3} = 60.6525896883197$$
$$x_{4} = 77.0838695174728$$
$$x_{5} = 100$$
$$x_{6} = 64.5261565642568$$
$$x_{7} = 6.04601858923138$$
$$x_{8} = 78$$
$$x_{9} = 70.3757428307212$$
$$x_{10} = 94.25$$
$$x_{11} = 74.3855050604014$$
$$x_{12} = 72.3229314979781$$
$$x_{13} = 51.1071218018637$$
$$x_{14} = 80$$
$$x_{15} = 98$$
$$x_{16} = 52.9946996224697$$
$$x_{17} = 62.5865072888081$$
$$x_{18} = 77.1862034519915$$
$$x_{19} = 96$$
$$x_{20} = 77.8203482568241$$
$$x_{21} = 77.2141744553657$$
$$x_{22} = 56.8055953511733$$
$$x_{23} = 77.3618108426466$$
$$x_{24} = 45.5503908345065$$
$$x_{25} = 54.8948687600961$$
$$x_{26} = 84$$
$$x_{27} = 77.0258369002429$$
$$x_{28} = 82$$
$$x_{29} = 40.2805348535545$$
$$x_{30} = 90$$
$$x_{31} = 47.3809756966331$$
$$x_{32} = 43.7493508424981$$
$$x_{33} = 66.4707529956174$$
$$x_{34} = 77.0566778668501$$
$$x_{35} = 68.419631664462$$
$$x_{36} = 77.1427592390923$$
$$x_{37} = 77.1877987770559$$
$$x_{38} = 86$$
$$x_{39} = 88$$
$$x_{40} = 58.7252674784738$$
$$x_{41} = 77.6251821417523$$
$$x_{42} = 92$$
$$x_{43} = 0$$
$$x_{44} = 41.9873266562552$$
$$x_{45} = 49.234743592684$$
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
$$\left[90, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$