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$$12 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(- \frac{\left(\tan^{2}{\left(2 x \right)} + 1\right) \sin^{2}{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)}}{\tan^{2}{\left(2 x \right)}} - \frac{\left(\tan^{2}{\left(2 x \right)} + 1\right) \sin{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)} \cos{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)}}{\tan^{2}{\left(2 x \right)}} + \frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \cos^{2}{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)}}{\tan^{2}{\left(2 x \right)}} + 2 \sin{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)} \cos{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)}\right) \sin{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 52.2289778659303$$
$$x_{2} = 61.9990653791785$$
$$x_{3} = 1.96349540849362$$
$$x_{4} = 83.990213954307$$
$$x_{5} = -4.31968989868597$$
$$x_{6} = 22.3838476568273$$
$$x_{7} = -78.1471172580461$$
$$x_{8} = 66.3661448070844$$
$$x_{9} = 94.4116768693319$$
$$x_{10} = -7.69008437233639$$
$$x_{11} = 80.2745099281778$$
$$x_{12} = 60.0829594999048$$
$$x_{13} = 40.0079168040499$$
$$x_{14} = 82.0741080750334$$
$$x_{15} = -71.8639319508665$$
$$x_{16} = -79.717913584841$$
$$x_{17} = 76.0197245879144$$
$$x_{18} = 8.24668071567321$$
$$x_{19} = -51.6723815225935$$
$$x_{20} = -86.0010988920206$$
$$x_{21} = -35.7356164345839$$
$$x_{22} = -93.8550805259951$$
$$x_{23} = -26.0820366537539$$
$$x_{24} = -92.0554823791395$$
$$x_{25} = 54.0285760127858$$
$$x_{26} = 98.0108731630429$$
$$x_{27} = -20.0276531666349$$
$$x_{28} = 36.2922127779207$$
$$x_{29} = 96.2112750161874$$
$$x_{30} = 67.9369411338793$$
$$x_{31} = 10.0462788625287$$
$$x_{32} = 88.3572933822129$$
$$x_{33} = 45.9457925587507$$
$$x_{34} = 23.9546439836222$$
$$x_{35} = -57.7267650097125$$
$$x_{36} = -64.009950316892$$
$$x_{37} = -42.0188017417635$$
$$x_{38} = -40.6768072350292$$
$$x_{39} = -73.663530097722$$
$$x_{40} = -27.8816348006094$$
$$x_{41} = 14.3010642027922$$
$$x_{42} = -13.7444678594553$$
$$x_{43} = 30.2378292908018$$
$$x_{44} = -32.5940237809941$$
$$x_{45} = -5.89048622548086$$
$$x_{46} = -21.8272513134905$$
$$x_{47} = 58.2833613530493$$
$$x_{48} = -29.6812329474649$$
$$x_{49} = 89.9280897090078$$
$$x_{50} = 44.3749962319558$$
$$x_{51} = -87.8006970388761$$
$$x_{52} = 32.0374274376573$$
$$x_{53} = 74.2201264410589$$
$$x_{54} = -49.872783375738$$
$$x_{55} = -48.0731852288824$$
$$x_{56} = 16.1006623496477$$
$$x_{57} = -100.138265833175$$
$$x_{58} = -70.064333804011$$
$$x_{59} = 38.0918109247762$$
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
$$\left[96.2112750161874, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -100.138265833175\right]$$