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$$\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{6 \tan{\left(x \right)}}{x^{2}}\right)}{x^{3}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 84.8583106190521$$
$$x_{2} = -66.0187938781021$$
$$x_{3} = -69.1583357737765$$
$$x_{4} = -62.8794552929189$$
$$x_{5} = -94.2795677093343$$
$$x_{6} = -56.6015217996642$$
$$x_{7} = -91.1390680936179$$
$$x_{8} = -25.2499214706009$$
$$x_{9} = -50.3248847443636$$
$$x_{10} = 3.69358314705748$$
$$x_{11} = 94.2795677093343$$
$$x_{12} = 19.0036683009869$$
$$x_{13} = 34.6434756494872$$
$$x_{14} = -84.8583106190521$$
$$x_{15} = 53.4630130760851$$
$$x_{16} = 12.7892677205421$$
$$x_{17} = 44.0500884228484$$
$$x_{18} = -15.8904615094706$$
$$x_{19} = -6.66644065689776$$
$$x_{20} = 37.7780284480501$$
$$x_{21} = 15.8904615094706$$
$$x_{22} = 97.420137568151$$
$$x_{23} = -78.5779395044499$$
$$x_{24} = 28.3788902944365$$
$$x_{25} = 9.70876728081698$$
$$x_{26} = -40.9136395730782$$
$$x_{27} = -59.7403517227637$$
$$x_{28} = -72.2980547201869$$
$$x_{29} = 22.1243392482168$$
$$x_{30} = -37.7780284480501$$
$$x_{31} = -28.3788902944365$$
$$x_{32} = -12.7892677205421$$
$$x_{33} = 59.7403517227637$$
$$x_{34} = 72.2980547201869$$
$$x_{35} = 91.1390680936179$$
$$x_{36} = -53.4630130760851$$
$$x_{37} = 87.9986462029283$$
$$x_{38} = -9.70876728081698$$
$$x_{39} = -3.69358314705748$$
$$x_{40} = -44.0500884228484$$
$$x_{41} = 66.0187938781021$$
$$x_{42} = 78.5779395044499$$
$$x_{43} = 6.66644065689776$$
$$x_{44} = 47.1872114097454$$
$$x_{45} = 25.2499214706009$$
$$x_{46} = 75.4379287840465$$
$$x_{47} = 40.9136395730782$$
$$x_{48} = -22.1243392482168$$
$$x_{49} = -47.1872114097454$$
$$x_{50} = -81.718071233071$$
$$x_{51} = -100.560771117475$$
$$x_{52} = -87.9986462029283$$
$$x_{53} = -75.4379287840465$$
$$x_{54} = 56.6015217996642$$
$$x_{55} = 50.3248847443636$$
$$x_{56} = 69.1583357737765$$
$$x_{57} = 100.560771117475$$
$$x_{58} = 31.5102856697475$$
$$x_{59} = -97.420137568151$$
$$x_{60} = 62.8794552929189$$
$$x_{61} = 81.718071233071$$
$$x_{62} = -34.6434756494872$$
$$x_{63} = -19.0036683009869$$
$$x_{64} = -31.5102856697475$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{6 \tan{\left(x \right)}}{x^{2}}\right)}{x^{3}}\right) = \infty$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{6 \tan{\left(x \right)}}{x^{2}}\right)}{x^{3}}\right) = \infty$$
- 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
$$\left[100.560771117475, \infty\right)$$
Convex at the intervals
$$\left[-3.69358314705748, 3.69358314705748\right]$$