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$$\left(- \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\tan{\left(x \right)} \right)} + 2 \cos{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -84.1679431384576$$
$$x_{2} = 62.8318530717959$$
$$x_{3} = -50.2654824574367$$
$$x_{4} = 18.194497413072$$
$$x_{5} = -85.4780601553912$$
$$x_{6} = 3.79665116205657$$
$$x_{7} = 72.2566310325652$$
$$x_{8} = -43.9822971502571$$
$$x_{9} = 58.2704602532356$$
$$x_{10} = -37.6991118430775$$
$$x_{11} = -13.2214291228259$$
$$x_{12} = 24.4776827202516$$
$$x_{13} = -94.2477796076938$$
$$x_{14} = -51.987274946056$$
$$x_{15} = -21.9911485751286$$
$$x_{16} = 50.2654824574367$$
$$x_{17} = -15.707963267949$$
$$x_{18} = 84.1679431384576$$
$$x_{19} = -33.9024606810209$$
$$x_{20} = 19.5046144300055$$
$$x_{21} = 21.9911485751286$$
$$x_{22} = -87.9645943005142$$
$$x_{23} = -72.2566310325652$$
$$x_{24} = 32.0709850443647$$
$$x_{25} = -24.4776827202516$$
$$x_{26} = 41.4957630051341$$
$$x_{27} = -77.8847578312781$$
$$x_{28} = -35.2125776979545$$
$$x_{29} = 36.279311678107$$
$$x_{30} = 54.0621336194933$$
$$x_{31} = -9.42477796076938$$
$$x_{32} = -3.79665116205657$$
$$x_{33} = -90.4511284456372$$
$$x_{34} = 85.4780601553912$$
$$x_{35} = 47.7789483123137$$
$$x_{36} = -55.8936092561495$$
$$x_{37} = -68.4599798705087$$
$$x_{38} = -62.1767945633291$$
$$x_{39} = -25.7877997371851$$
$$x_{40} = -46.4688312953801$$
$$x_{41} = -99.8759064064066$$
$$x_{42} = 8.7697194523026$$
$$x_{43} = -65.9734457253857$$
$$x_{44} = 15.707963267949$$
$$x_{45} = -91.7612454625708$$
$$x_{46} = 28.2743338823081$$
$$x_{47} = 25.7877997371851$$
$$x_{48} = 94.2477796076938$$
$$x_{49} = -11.9113121058924$$
$$x_{50} = 64.1328582855002$$
$$x_{51} = -2.48653414512302$$
$$x_{52} = 38.3541703515443$$
$$x_{53} = 6.28318530717959$$
$$x_{54} = -74.0972184724507$$
$$x_{55} = -47.7789483123137$$
$$x_{56} = 62.1767945633291$$
$$x_{57} = -53.4070751110265$$
$$x_{58} = 76.0532821946218$$
$$x_{59} = 14.4069580542446$$
$$x_{60} = -5.62812679871281$$
$$x_{61} = -75.398223686155$$
$$x_{62} = 2.48653414512302$$
$$x_{63} = -96.0883670475793$$
$$x_{64} = 98.0444307697504$$
$$x_{65} = 30.7608680274312$$
$$x_{66} = 0$$
$$x_{67} = -7.7504834037219$$
$$x_{68} = 46.4688312953801$$
$$x_{69} = -40.1856459882005$$
$$x_{70} = 69.7700968874422$$
$$x_{71} = -97.3893722612836$$
$$x_{72} = -18.194497413072$$
$$x_{73} = 52.7520166025597$$
$$x_{74} = 68.4599798705087$$
$$x_{75} = -28.2743338823081$$
$$x_{76} = 74.7431651776883$$
$$x_{77} = 100.530964914873$$
$$x_{78} = -19.5046144300055$$
$$x_{79} = 40.1856459882005$$
$$x_{80} = -69.7700968874422$$
$$x_{81} = 96.7343137528168$$
$$x_{82} = 80.2616088283641$$
$$x_{83} = -41.4957630051341$$
$$x_{84} = 78.5398163397448$$
$$x_{85} = 91.7612454625708$$
$$x_{86} = -57.2037262730831$$
$$x_{87} = 90.4511284456372$$
$$x_{88} = 87.9645943005142$$
$$x_{89} = 43.9822971502571$$
$$x_{90} = -31.4159265358979$$
$$x_{91} = -81.6814089933346$$
$$x_{92} = 56.5486677646163$$
$$x_{93} = -59.6902604182061$$
$$x_{94} = 65.9734457253857$$
$$x_{95} = -63.4869115802626$$
$$x_{96} = -79.1948748482116$$
$$x_{97} = -29.9961263709274$$
$$x_{98} = 10.0798364692362$$
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
$$\left[100.530964914873, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -97.3893722612836\right]$$