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$$2 \left(\left(x - 1\right) \left(\tan^{2}{\left(x + 1 \right)} + 1\right) \tan{\left(x + 1 \right)} + \tan^{2}{\left(x + 1 \right)} + 1\right) = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 86.9529605548182$$
$$x_{2} = 46.1017213333012$$
$$x_{3} = -29.2412785419537$$
$$x_{4} = -32.385982798361$$
$$x_{5} = 33.5267849771474$$
$$x_{6} = 64.9578116926087$$
$$x_{7} = 24.0894584701575$$
$$x_{8} = -44.9605427891551$$
$$x_{9} = -57.5315846337464$$
$$x_{10} = -19.8015194924769$$
$$x_{11} = -35.5301513773073$$
$$x_{12} = 52.3876176260311$$
$$x_{13} = -54.3890229534413$$
$$x_{14} = -73.2431625932093$$
$$x_{15} = 74.3845976918532$$
$$x_{16} = 20.9410427201664$$
$$x_{17} = -85.8114829429597$$
$$x_{18} = 14.6347523490529$$
$$x_{19} = -13.4975022995969$$
$$x_{20} = -3.94193875707095$$
$$x_{21} = 30.3819051153235$$
$$x_{22} = -7.16126299361986$$
$$x_{23} = 42.9584685702391$$
$$x_{24} = 36.6710852734038$$
$$x_{25} = 99.5208151240999$$
$$x_{26} = -10.3367973396921$$
$$x_{27} = -79.527398844386$$
$$x_{28} = 58.6729229962298$$
$$x_{29} = -98.3793101442334$$
$$x_{30} = 55.5303314021238$$
$$x_{31} = -88.9534779008869$$
$$x_{32} = -95.2373890097998$$
$$x_{33} = 39.8149469233971$$
$$x_{34} = -82.6694577698462$$
$$x_{35} = -38.6739116985542$$
$$x_{36} = 80.6688576964204$$
$$x_{37} = -66.9587319747743$$
$$x_{38} = -0.369187240041531$$
$$x_{39} = -16.6513709380068$$
$$x_{40} = 5.04057057804068$$
$$x_{41} = 17.7900671729686$$
$$x_{42} = 71.242395577731$$
$$x_{43} = 49.2447577845162$$
$$x_{44} = 93.2369383889823$$
$$x_{45} = -41.8173537252262$$
$$x_{46} = 8.2884254693857$$
$$x_{47} = -92.0954457031868$$
$$x_{48} = -70.1009748014528$$
$$x_{49} = 83.8109265327022$$
$$x_{50} = -76.3853020548292$$
$$x_{51} = 61.8154113530589$$
$$x_{52} = -26.0958519530331$$
$$x_{53} = 90.0949634463778$$
$$x_{54} = -63.8164261078551$$
$$x_{55} = 27.2362371018485$$
$$x_{56} = -51.2463446992226$$
$$x_{57} = 96.3788881449316$$
$$x_{58} = 11.4711589550628$$
$$x_{59} = -48.1035274828611$$
$$x_{60} = 68.1001363828001$$
$$x_{61} = -60.6740475633817$$
$$x_{62} = -22.9494181476817$$
$$x_{63} = 77.5267497572666$$
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
$$\left[99.5208151240999, \infty\right)$$
Convex at the intervals
$$\left[-0.369187240041531, 5.04057057804068\right]$$