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$$\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{x} + \frac{\tan{\left(x \right)}}{x^{2}}\right)}{x} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -40.8651557120368$$
$$x_{2} = 28.3095989977492$$
$$x_{3} = 97.399637797279$$
$$x_{4} = 34.5864001254547$$
$$x_{5} = 18.9022635301866$$
$$x_{6} = 62.8477591701485$$
$$x_{7} = 69.1294999494455$$
$$x_{8} = -53.4257839289366$$
$$x_{9} = -22.0364040421205$$
$$x_{10} = 65.9885952215714$$
$$x_{11} = 87.9759590846368$$
$$x_{12} = 56.5663387618027$$
$$x_{13} = 50.2853584856195$$
$$x_{14} = -72.2704644131198$$
$$x_{15} = -87.9759590846368$$
$$x_{16} = 47.1450882107807$$
$$x_{17} = 3.40690770689403$$
$$x_{18} = 81.6936474025491$$
$$x_{19} = -28.3095989977492$$
$$x_{20} = -31.4476826131772$$
$$x_{21} = 31.4476826131772$$
$$x_{22} = 6.43387606467487$$
$$x_{23} = 84.8347870810872$$
$$x_{24} = -75.4114811587071$$
$$x_{25} = 22.0364040421205$$
$$x_{26} = -3.40690770689403$$
$$x_{27} = -15.7710339845247$$
$$x_{28} = 59.7070026124805$$
$$x_{29} = -25.1723840259432$$
$$x_{30} = -50.2853584856195$$
$$x_{31} = -12.6448047030571$$
$$x_{32} = 72.2704644131198$$
$$x_{33} = 37.7255942551796$$
$$x_{34} = -100.540909803285$$
$$x_{35} = -81.6936474025491$$
$$x_{36} = 15.7710339845247$$
$$x_{37} = -69.1294999494455$$
$$x_{38} = -78.5525439224845$$
$$x_{39} = 75.4114811587071$$
$$x_{40} = -34.5864001254547$$
$$x_{41} = 9.52822730114541$$
$$x_{42} = 25.1723840259432$$
$$x_{43} = -6.43387606467487$$
$$x_{44} = 53.4257839289366$$
$$x_{45} = -37.7255942551796$$
$$x_{46} = 78.5525439224845$$
$$x_{47} = -97.399637797279$$
$$x_{48} = -44.0050062097373$$
$$x_{49} = -18.9022635301866$$
$$x_{50} = 44.0050062097373$$
$$x_{51} = -62.8477591701485$$
$$x_{52} = -91.1171600731987$$
$$x_{53} = 91.1171600731987$$
$$x_{54} = -56.5663387618027$$
$$x_{55} = 100.540909803285$$
$$x_{56} = -94.2583871514792$$
$$x_{57} = -47.1450882107807$$
$$x_{58} = 40.8651557120368$$
$$x_{59} = -59.7070026124805$$
$$x_{60} = -65.9885952215714$$
$$x_{61} = -84.8347870810872$$
$$x_{62} = 94.2583871514792$$
$$x_{63} = -9.52822730114541$$
$$x_{64} = 12.6448047030571$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{x} + \frac{\tan{\left(x \right)}}{x^{2}}\right)}{x}\right) = \frac{2}{3}$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{x} + \frac{\tan{\left(x \right)}}{x^{2}}\right)}{x}\right) = \frac{2}{3}$$
- limits are equal, then skip the corresponding 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
$$\left[100.540909803285, \infty\right)$$
Convex at the intervals
$$\left[-3.40690770689403, 3.40690770689403\right]$$