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$$4 \left(2 x \left(2 x - 1\right) \left(\cot^{2}{\left(2 x \right)} + 1\right) \cot{\left(2 x \right)} - \left(4 x - 1\right) \left(\cot^{2}{\left(2 x \right)} + 1\right) + \cot{\left(2 x \right)}\right) = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -60.467424876252$$
$$x_{2} = -27.4709109980685$$
$$x_{3} = -69.8933090331141$$
$$x_{4} = -2.16437979583232$$
$$x_{5} = 55.7542628064408$$
$$x_{6} = -21.1824476760206$$
$$x_{7} = 74.6061017583645$$
$$x_{8} = -10.162384360162$$
$$x_{9} = 19.6091622908847$$
$$x_{10} = 41.6140185364853$$
$$x_{11} = -54.183289332862$$
$$x_{12} = 3.79112174721024$$
$$x_{13} = -11.7394187725236$$
$$x_{14} = 40.0427453584178$$
$$x_{15} = -30.6143372910994$$
$$x_{16} = 82.4607256938572$$
$$x_{17} = -47.8989058469511$$
$$x_{18} = -19.609810749281$$
$$x_{19} = 18.0360920009685$$
$$x_{20} = -79.3189311602732$$
$$x_{21} = -16.4635010190137$$
$$x_{22} = 77.7479669535723$$
$$x_{23} = -35.3288698441239$$
$$x_{24} = -13.3150128515235$$
$$x_{25} = -93.4570459826742$$
$$x_{26} = -63.6094225367516$$
$$x_{27} = -77.7480083049387$$
$$x_{28} = 54.1832042068874$$
$$x_{29} = -38.4716017991362$$
$$x_{30} = -43.185390812245$$
$$x_{31} = -76.1770802490528$$
$$x_{32} = 33.7572057770887$$
$$x_{33} = 16.4625819979758$$
$$x_{34} = -3.80737870526536$$
$$x_{35} = 36.9000764281067$$
$$x_{36} = 66.7513261269453$$
$$x_{37} = -41.6141628173047$$
$$x_{38} = -84.0316714304799$$
$$x_{39} = 22.7543516043807$$
$$x_{40} = 44.7564639424648$$
$$x_{41} = 85.6025421727294$$
$$x_{42} = -18.0368581393357$$
$$x_{43} = -25.8990327632379$$
$$x_{44} = 60.4673565199096$$
$$x_{45} = -82.4607624543778$$
$$x_{46} = -62.038428812516$$
$$x_{47} = -40.0429011772707$$
$$x_{48} = 25.8986606004142$$
$$x_{49} = -74.6061466652612$$
$$x_{50} = 5.40256448970248$$
$$x_{51} = 27.4705801530128$$
$$x_{52} = -98.1696904135897$$
$$x_{53} = 98.1696644753665$$
$$x_{54} = 8.57979670117484$$
$$x_{55} = -91.8861586993066$$
$$x_{56} = -52.6122201588358$$
$$x_{57} = 69.8932578672581$$
$$x_{58} = 52.6121298748597$$
$$x_{59} = 93.4570173628919$$
$$x_{60} = -85.6025762847351$$
$$x_{61} = 84.0316360313412$$
$$x_{62} = 76.1770371748959$$
$$x_{63} = 80.8898109150899$$
$$x_{64} = 11.7376174464405$$
$$x_{65} = 32.1856762132309$$
$$x_{66} = 91.8861290927073$$
$$x_{67} = 96.5987849099627$$
$$x_{68} = -90.3152682609883$$
$$x_{69} = -32.1859173107788$$
$$x_{70} = -46.3277594789849$$
$$x_{71} = 99.7405414125112$$
$$x_{72} = 24.3265946262001$$
$$x_{73} = -5.41084155066206$$
$$x_{74} = 68.3222959089102$$
$$x_{75} = 88.7443427609323$$
$$x_{76} = -49.4700301023807$$
$$x_{77} = -55.7543432041321$$
$$x_{78} = 46.3276430512942$$
$$x_{79} = -71.464261461156$$
$$x_{80} = 58.896337863845$$
$$x_{81} = -33.7574249685089$$
$$x_{82} = -87.1734772393062$$
$$x_{83} = -8.58314655257581$$
$$x_{84} = 10.1599860824752$$
$$x_{85} = 71.4642125196339$$
$$x_{86} = -68.3223494542592$$
$$x_{87} = -65.1804067843426$$
$$x_{88} = 90.3152376156395$$
$$x_{89} = -96.5988116985632$$
$$x_{90} = -99.7405665402152$$
$$x_{91} = -57.3253830249138$$
$$x_{92} = -24.3270163592699$$
$$x_{93} = 14.8884911015381$$
$$x_{94} = 49.4699279898137$$
$$x_{95} = 30.6140708315123$$
$$x_{96} = 62.0383638734434$$
$$x_{97} = 2.11828268940796$$
$$x_{98} = 38.4714330007798$$
$$x_{99} = 63.6093607649561$$
$$x_{100} = 47.8987969287076$$
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
$$\left(-\infty, -99.7405665402152\right]$$
Convex at the intervals
$$\left[99.7405414125112, \infty\right)$$