Graphing y = arctan(1/x)/(x+2)

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
$$\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 4266.65367445192$$
$$x_{2} = 9937.37395136824$$
$$x_{3} = 3612.17008848393$$
$$x_{4} = 9719.28819648547$$
$$x_{5} = -4452.88432742144$$
$$x_{6} = -10559.8323524182$$
$$x_{7} = 10155.4590528456$$
$$x_{8} = 5139.19521736223$$
$$x_{9} = -4671.03531898808$$
$$x_{10} = -1833.45731857528$$
$$x_{11} = 7320.28554933361$$
$$x_{12} = -4889.17887858538$$
$$x_{13} = -3143.73181577214$$
$$x_{14} = 5793.55185085321$$
$$x_{15} = -2488.85074573824$$
$$x_{16} = -6197.92913031119$$
$$x_{17} = -8597.0397249602$$
$$x_{18} = 8192.66534690497$$
$$x_{19} = 7974.57237255382$$
$$x_{20} = 6229.77317138162$$
$$x_{21} = -9905.57533377917$$
$$x_{22} = -5325.44745340199$$
$$x_{23} = 10809.710824617$$
$$x_{24} = -5543.57396632486$$
$$x_{25} = -4234.72471680378$$
$$x_{26} = 11027.7936852099$$
$$x_{27} = -5107.31598521205$$
$$x_{28} = 2084.45236245165$$
$$x_{29} = 2957.5753803023$$
$$x_{30} = 2739.34003587$$
$$x_{31} = 1866.06649845331$$
$$x_{32} = -7942.75750807917$$
$$x_{33} = -8378.94691851695$$
$$x_{34} = -9687.48824590579$$
$$x_{35} = 7538.38258474053$$
$$x_{36} = 9283.11454648586$$
$$x_{37} = 8410.75717085281$$
$$x_{38} = 4484.79808915367$$
$$x_{39} = -5761.69610121046$$
$$x_{40} = -10123.6616831784$$
$$x_{41} = -2707.18224487169$$
$$x_{42} = -9469.4003676517$$
$$x_{43} = -4016.55503372661$$
$$x_{44} = 3830.34101271359$$
$$x_{45} = 6665.98425095459$$
$$x_{46} = 7102.18691631106$$
$$x_{47} = 3175.78984726872$$
$$x_{48} = -9251.31164213201$$
$$x_{49} = 10591.6274538209$$
$$x_{50} = 10373.5435412909$$
$$x_{51} = 2302.78597024282$$
$$x_{52} = 5357.3178943352$$
$$x_{53} = 4048.50171707908$$
$$x_{54} = 6884.08653869763$$
$$x_{55} = -2052.01060774317$$
$$x_{56} = 9065.02655079485$$
$$x_{57} = -10341.7473415644$$
$$x_{58} = -7288.46215665169$$
$$x_{59} = -5979.81434909453$$
$$x_{60} = -6852.25605773224$$
$$x_{61} = -7070.36014300167$$
$$x_{62} = -2925.47299918569$$
$$x_{63} = -8815.13139310588$$
$$x_{64} = -7724.66068520671$$
$$x_{65} = 4921.06809983418$$
$$x_{66} = -7506.56228301616$$
$$x_{67} = 7756.47815347094$$
$$x_{68} = -3580.17780168701$$
$$x_{69} = -10777.9167556495$$
$$x_{70} = -3798.37347953912$$
$$x_{71} = 3393.98712459383$$
$$x_{72} = 8846.93769953506$$
$$x_{73} = 6447.87986596005$$
$$x_{74} = -6416.04080676295$$
$$x_{75} = -1614.76423318349$$
$$x_{76} = 2521.07901482993$$
$$x_{77} = 5575.43662977994$$
$$x_{78} = -8160.85288071644$$
$$x_{79} = 9501.20174420201$$
$$x_{80} = 4702.93595515051$$
$$x_{81} = 1647.6121150313$$
$$x_{82} = -9033.22200687244$$
$$x_{83} = -2270.4661524954$$
$$x_{84} = -3361.96514138765$$
$$x_{85} = -6634.14969172341$$
$$x_{86} = 8628.84792943952$$
$$x_{87} = 6011.6639252941$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -2$$
$$x_{2} = 0$$

$$\lim_{x \to -2^-}\left(\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2}\right) = \infty$$
$$\lim_{x \to -2^+}\left(\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2}\right) = -\infty$$
- the limits are not equal, so
$$x_{1} = -2$$
- is an inflection point
$$\lim_{x \to 0^-}\left(\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2}\right) = \frac{1}{2} - \frac{\pi}{8}$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2}\right) = \frac{\pi}{8} + \frac{1}{2}$$
- the limits are not equal, so
$$x_{2} = 0$$
- 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:
Have no bends at the whole real axis