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$$x \left(\tan^{2}{\left(x \right)} + 1\right) + x \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 37.6460725978858$$
$$x_{2} = 69.086103133906$$
$$x_{3} = -56.513303680917$$
$$x_{4} = 56.513303680917$$
$$x_{5} = -91.0842354292587$$
$$x_{6} = 18.7435508863884$$
$$x_{7} = 91.0842354292587$$
$$x_{8} = -18.7435508863884$$
$$x_{9} = 28.2036276766186$$
$$x_{10} = -40.7917435045268$$
$$x_{11} = -31.3522859596756$$
$$x_{12} = 31.3522859596756$$
$$x_{13} = -65.9431328173515$$
$$x_{14} = -78.5143529238898$$
$$x_{15} = -53.3696312584227$$
$$x_{16} = -37.6460725978858$$
$$x_{17} = -84.7994242285037$$
$$x_{18} = 0$$
$$x_{19} = 53.3696312584227$$
$$x_{20} = 34.4996607566446$$
$$x_{21} = 2.51787226577809$$
$$x_{22} = -5.96726435810305$$
$$x_{23} = 100.511071202847$$
$$x_{24} = -72.2289536776301$$
$$x_{25} = 62.8000247676753$$
$$x_{26} = -59.6567572450692$$
$$x_{27} = 59.6567572450692$$
$$x_{28} = -97.3688368609732$$
$$x_{29} = -28.2036276766186$$
$$x_{30} = 47.0814548431779$$
$$x_{31} = 43.93683212641$$
$$x_{32} = 72.2289536776301$$
$$x_{33} = -100.511071202847$$
$$x_{34} = 97.3688368609732$$
$$x_{35} = 5.96726435810305$$
$$x_{36} = 21.9002649847656$$
$$x_{37} = 84.7994242285037$$
$$x_{38} = 12.4075419598293$$
$$x_{39} = -34.4996607566446$$
$$x_{40} = 75.3716994163885$$
$$x_{41} = -81.6569248399483$$
$$x_{42} = 40.7917435045268$$
$$x_{43} = 9.21332735720748$$
$$x_{44} = -15.5808081405648$$
$$x_{45} = -50.2256989613876$$
$$x_{46} = -43.93683212641$$
$$x_{47} = 94.2265597445368$$
$$x_{48} = 78.5143529238898$$
$$x_{49} = 87.9418588589466$$
$$x_{50} = -75.3716994163885$$
$$x_{51} = -25.0532054465023$$
$$x_{52} = 50.2256989613876$$
$$x_{53} = -87.9418588589466$$
$$x_{54} = 15.5808081405648$$
$$x_{55} = 25.0532054465023$$
$$x_{56} = -2.51787226577809$$
$$x_{57} = -94.2265597445368$$
$$x_{58} = -62.8000247676753$$
$$x_{59} = 81.6569248399483$$
$$x_{60} = 65.9431328173515$$
$$x_{61} = -21.9002649847656$$
$$x_{62} = -47.0814548431779$$
$$x_{63} = -12.4075419598293$$
$$x_{64} = -69.086103133906$$
$$x_{65} = -9.21332735720748$$
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
$$\left[100.511071202847, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.511071202847\right]$$