Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada$$2 \left(4 x^{2} \left(\cot^{2}{\left(x^{2} \right)} + 1\right) \cot{\left(x^{2} \right)} - \cot^{2}{\left(x^{2} \right)} - 1\right) = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 45.9985057804882$$
$$x_{2} = -9.78856305960857$$
$$x_{3} = 72.2478798723152$$
$$x_{4} = -67.9453482609181$$
$$x_{5} = -38.893144607161$$
$$x_{6} = 2.15842031656434$$
$$x_{7} = 34.254702495525$$
$$x_{8} = -87.9376499827146$$
$$x_{9} = -97.7504681816382$$
$$x_{10} = 80.0062184622719$$
$$x_{11} = 20.2478944955663$$
$$x_{12} = 12.3436462566901$$
$$x_{13} = -75.7503725285458$$
$$x_{14} = -91.7491007345316$$
$$x_{15} = -3.75758735559995$$
$$x_{16} = -51.7514023369347$$
$$x_{17} = -57.7477317893834$$
$$x_{18} = -14.0124326083036$$
$$x_{19} = 90.1428276001121$$
$$x_{20} = 88.2229888812924$$
$$x_{21} = 96.0320009554325$$
$$x_{22} = 62.2507387401102$$
$$x_{23} = -85.7490383137135$$
$$x_{24} = 40.1256287139733$$
$$x_{25} = -77.977897765382$$
$$x_{26} = 16.0012132306729$$
$$x_{27} = 42.0000492055384$$
$$x_{28} = -45.7588394107798$$
$$x_{29} = 32.173817582749$$
$$x_{30} = -5.74274694367216$$
$$x_{31} = 74.2211740185213$$
$$x_{32} = 58.019104216233$$
$$x_{33} = 54.1251795760043$$
$$x_{34} = 4.15503066662393$$
$$x_{35} = 37.7869512658615$$
$$x_{36} = -95.7535280406693$$
$$x_{37} = -71.7460766477137$$
$$x_{38} = -55.754872883948$$
$$x_{39} = -40.0080154571131$$
$$x_{40} = 82.4619746424392$$
$$x_{41} = 48.0037300937962$$
$$x_{42} = 51.9936571702341$$
$$x_{43} = 30.0011002714247$$
$$x_{44} = -42.0000492055384$$
$$x_{45} = 68.2452273442545$$
$$x_{46} = 65.9986223856813$$
$$x_{47} = -8.02488425811924$$
$$x_{48} = -23.7469037558692$$
$$x_{49} = -42.8516133870942$$
$$x_{50} = -47.83983840802$$
$$x_{51} = -65.9986223856813$$
$$x_{52} = -21.7441754410407$$
$$x_{53} = 99.9905925327219$$
$$x_{54} = 51.0177363026731$$
$$x_{55} = -35.9987052085305$$
$$x_{56} = 24.0100357823293$$
$$x_{57} = 35.955043923882$$
$$x_{58} = -17.768772323048$$
$$x_{59} = 84.2521724679374$$
$$x_{60} = 26.259840534989$$
$$x_{61} = -53.7757933570406$$
$$x_{62} = 94.2488896480008$$
$$x_{63} = -20.0138062145804$$
$$x_{64} = -79.7505754705589$$
$$x_{65} = -90.0033145301847$$
$$x_{66} = 56.2038373604716$$
$$x_{67} = -81.8693342414289$$
$$x_{68} = 92.1249831707776$$
$$x_{69} = 27.9969114032736$$
$$x_{70} = 8.2183052138805$$
$$x_{71} = -69.770321522697$$
$$x_{72} = -83.7472719764304$$
$$x_{73} = 6.26606264138994$$
$$x_{74} = -73.7541024874687$$
$$x_{75} = -26.0194705476381$$
$$x_{76} = 63.9927024191448$$
$$x_{77} = 76.2464239323044$$
$$x_{78} = 86.0051153657684$$
$$x_{79} = 69.8603187519718$$
$$x_{80} = 10.2587025689995$$
$$x_{81} = -94.3987687094098$$
$$x_{82} = -2.15842031656434$$
$$x_{83} = -63.7467649053452$$
$$x_{84} = 60.25039620081$$
$$x_{85} = -33.7465126702331$$
$$x_{86} = -62.2255002469528$$
$$x_{87} = 77.9980392867871$$
$$x_{88} = -31.7313755955179$$
$$x_{89} = -50.1795417731822$$
$$x_{90} = -59.752991396122$$
$$x_{91} = -27.9969114032736$$
$$x_{92} = 98.2473582456787$$
$$x_{93} = -12.0864530375985$$
$$x_{94} = -16.0012132306729$$
$$x_{95} = -99.8176391013562$$
$$x_{96} = -29.896200933167$$
$$x_{97} = 18.0320290256093$$
$$x_{98} = 14.1240886540409$$
$$x_{99} = 22.2441079319026$$
$$x_{100} = 45.032187761776$$
Intervalos de convexidad y concavidad:Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[-2.15842031656434, 2.15842031656434\right]$$
Convexa en los intervalos
$$\left(-\infty, -99.8176391013562\right]$$