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$$\frac{9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{x} + \frac{\tan{\left(3 x \right)}}{x^{2}}}{x} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -37.7020586193797$$
$$x_{2} = 59.6921217440971$$
$$x_{3} = -65.9751298043699$$
$$x_{4} = -28.2782623603624$$
$$x_{5} = -77.4940525549667$$
$$x_{6} = 76.4468746426753$$
$$x_{7} = 19.9023342041602$$
$$x_{8} = -79.5884099297947$$
$$x_{9} = 100.532070129183$$
$$x_{10} = 30.3723866910662$$
$$x_{11} = 61.7864537594165$$
$$x_{12} = -9.43653299924975$$
$$x_{13} = 41.890554238415$$
$$x_{14} = 26.1841813074948$$
$$x_{15} = -26.1841813074948$$
$$x_{16} = -39.7962986804533$$
$$x_{17} = 21.9961984071905$$
$$x_{18} = -99.4848842112145$$
$$x_{19} = -92.1545901877593$$
$$x_{20} = -41.890554238415$$
$$x_{21} = 48.1733936907667$$
$$x_{22} = -55.5034719939252$$
$$x_{23} = -52.3619994250983$$
$$x_{24} = 46.0791033967769$$
$$x_{25} = -57.59779431589$$
$$x_{26} = 52.3619994250983$$
$$x_{27} = 15.7150294035936$$
$$x_{28} = 24.0901548043733$$
$$x_{29} = -72.2581686851326$$
$$x_{30} = -33.513636601095$$
$$x_{31} = 10.4825608710591$$
$$x_{32} = -85.8714930952558$$
$$x_{33} = -8.39079467531438$$
$$x_{34} = -50.267692715982$$
$$x_{35} = -46.0791033967769$$
$$x_{36} = 6.30075451006221$$
$$x_{37} = 17.8085946429789$$
$$x_{38} = -75.3996972758354$$
$$x_{39} = 80.6355893520718$$
$$x_{40} = 68.0694730948384$$
$$x_{41} = -24.0901548043733$$
$$x_{42} = 83.7771303379305$$
$$x_{43} = 56.5506324811787$$
$$x_{44} = 37.7020586193797$$
$$x_{45} = -4.21493490101902$$
$$x_{46} = -31.4194623838264$$
$$x_{47} = 28.2782623603624$$
$$x_{48} = -96.343327974564$$
$$x_{49} = 39.7962986804533$$
$$x_{50} = 94.2489584987906$$
$$x_{51} = -13.6217185706789$$
$$x_{52} = 4.21493490101902$$
$$x_{53} = 12.5751980850599$$
$$x_{54} = -21.9961984071905$$
$$x_{55} = -94.2489584987906$$
$$x_{56} = -70.1638194780521$$
$$x_{57} = 81.6827692391381$$
$$x_{58} = -68.0694730948384$$
$$x_{59} = 8.39079467531438$$
$$x_{60} = -74.3525204774901$$
$$x_{61} = -35.6078367878772$$
$$x_{62} = -2.14462535489162$$
$$x_{63} = -53.4091553787305$$
$$x_{64} = 72.2581686851326$$
$$x_{65} = -30.3723866910662$$
$$x_{66} = 43.9848230807054$$
$$x_{67} = -83.7771303379305$$
$$x_{68} = 92.1545901877593$$
$$x_{69} = 65.9751298043699$$
$$x_{70} = 63.8807899107774$$
$$x_{71} = -97.3905131257488$$
$$x_{72} = 70.1638194780521$$
$$x_{73} = -48.1733936907667$$
$$x_{74} = 2.14462535489162$$
$$x_{75} = 98.4376985407423$$
$$x_{76} = 14.6683354032458$$
$$x_{77} = 78.5412309908953$$
$$x_{78} = 34.5607337431424$$
$$x_{79} = -87.9658573926638$$
$$x_{80} = -81.6827692391381$$
$$x_{81} = 90.060223122721$$
$$x_{82} = 87.9658573926638$$
$$x_{83} = 74.3525204774901$$
$$x_{84} = 96.343327974564$$
$$x_{85} = -19.9023342041602$$
$$x_{86} = -43.9848230807054$$
$$x_{87} = -17.8085946429789$$
$$x_{88} = 58.6449574258964$$
$$x_{89} = -63.8807899107774$$
$$x_{90} = -59.6921217440971$$
$$x_{91} = -15.7150294035936$$
$$x_{92} = 85.8714930952558$$
$$x_{93} = -61.7864537594165$$
$$x_{94} = -11.5288000418182$$
$$x_{95} = 36.654945229763$$
$$x_{96} = 54.4563129318412$$
$$x_{97} = -6.30075451006221$$
$$x_{98} = 50.267692715982$$
$$x_{99} = 32.4665459324263$$
$$x_{100} = -90.060223122721$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{x} + \frac{\tan{\left(3 x \right)}}{x^{2}}}{x}\right) = 9$$
$$\lim_{x \to 0^+}\left(\frac{9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} - \frac{3 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{x} + \frac{\tan{\left(3 x \right)}}{x^{2}}}{x}\right) = 9$$
- los límites son iguales, es decir omitimos el punto correspondiente
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
$$\left[100.532070129183, \infty\right)$$
Convexa en los intervalos
$$\left[-2.14462535489162, 2.14462535489162\right]$$