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$$\frac{2 \left(4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} - \frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right)}{x} + \frac{\tan{\left(2 x \right)}}{x^{2}}\right)}{x} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -50.2704549016424$$
$$x_{2} = 29.8535013062402$$
$$x_{3} = -73.8308132724072$$
$$x_{4} = -45.5585800365993$$
$$x_{5} = -1.70345385344701$$
$$x_{6} = -34.5647499747227$$
$$x_{7} = -83.2552079908878$$
$$x_{8} = -72.26009053615$$
$$x_{9} = -81.6844693977618$$
$$x_{10} = -56.5530879290695$$
$$x_{11} = 34.5647499747227$$
$$x_{12} = -37.7057405793536$$
$$x_{13} = -15.7238413065886$$
$$x_{14} = 53.4117551818158$$
$$x_{15} = 51.8411006147136$$
$$x_{16} = 80.1137330680957$$
$$x_{17} = 23.5725441053903$$
$$x_{18} = -53.4117551818158$$
$$x_{19} = 58.1237648337733$$
$$x_{20} = -47.1291935757396$$
$$x_{21} = 45.5585800365993$$
$$x_{22} = -58.1237648337733$$
$$x_{23} = 73.8308132724072$$
$$x_{24} = -95.8211848661661$$
$$x_{25} = -14.1547994988746$$
$$x_{26} = 89.5381826161457$$
$$x_{27} = 42.4173935405436$$
$$x_{28} = 64.4065306806787$$
$$x_{29} = 1.70345385344701$$
$$x_{30} = -78.5429991376474$$
$$x_{31} = 6.32240235152853$$
$$x_{32} = 75.401539073973$$
$$x_{33} = 50.2704549016424$$
$$x_{34} = -80.1137330680957$$
$$x_{35} = 78.5429991376474$$
$$x_{36} = -65.9772346210581$$
$$x_{37} = -17.2932000627273$$
$$x_{38} = 18.8627971275898$$
$$x_{39} = 72.26009053615$$
$$x_{40} = -31.4238795850742$$
$$x_{41} = -89.5381826161457$$
$$x_{42} = 40.8468237012746$$
$$x_{43} = -39.2762719612422$$
$$x_{44} = 100.53345156734$$
$$x_{45} = -3.21693803233743$$
$$x_{46} = 62.8358313576226$$
$$x_{47} = -61.2651370068557$$
$$x_{48} = -20.4325778560184$$
$$x_{49} = 81.6844693977618$$
$$x_{50} = 9.45113176509331$$
$$x_{51} = -36.1352322065599$$
$$x_{52} = -28.2831693809014$$
$$x_{53} = 43.9879795423184$$
$$x_{54} = 70.6893710694627$$
$$x_{55} = 87.9674361388446$$
$$x_{56} = 92.6796806391247$$
$$x_{57} = -87.9674361388446$$
$$x_{58} = -67.5479428567926$$
$$x_{59} = 48.6998188986395$$
$$x_{60} = 59.69444802068$$
$$x_{61} = 26.7128919644683$$
$$x_{62} = -7.88551699226233$$
$$x_{63} = -86.396691473835$$
$$x_{64} = 94.2504320159027$$
$$x_{65} = 28.2831693809014$$
$$x_{66} = 20.4325778560184$$
$$x_{67} = 97.3919391186353$$
$$x_{68} = -29.8535013062402$$
$$x_{69} = 84.825948721768$$
$$x_{70} = 36.1352322065599$$
$$x_{71} = 86.396691473835$$
$$x_{72} = 12.5861920129716$$
$$x_{73} = -22.0025031048686$$
$$x_{74} = 56.5530879290695$$
$$x_{75} = -97.3919391186353$$
$$x_{76} = -64.4065306806787$$
$$x_{77} = -51.8411006147136$$
$$x_{78} = 14.1547994988746$$
$$x_{79} = 31.4238795850742$$
$$x_{80} = 67.5479428567926$$
$$x_{81} = 15.7238413065886$$
$$x_{82} = -43.9879795423184$$
$$x_{83} = -100.53345156734$$
$$x_{84} = -59.69444802068$$
$$x_{85} = -25.1426792428098$$
$$x_{86} = 65.9772346210581$$
$$x_{87} = 95.8211848661661$$
$$x_{88} = -91.108930812026$$
$$x_{89} = -69.1186550951654$$
$$x_{90} = 22.0025031048686$$
$$x_{91} = -12.5861920129716$$
$$x_{92} = -94.2504320159027$$
$$x_{93} = 7.88551699226233$$
$$x_{94} = -9.45113176509331$$
$$x_{95} = -23.5725441053903$$
$$x_{96} = -75.401539073973$$
$$x_{97} = -6.32240235152853$$
$$x_{98} = 4.7641136505727$$
$$x_{99} = 37.7057405793536$$
$$x_{100} = -42.4173935405436$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} - \frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right)}{x} + \frac{\tan{\left(2 x \right)}}{x^{2}}\right)}{x}\right) = \frac{16}{3}$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} - \frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right)}{x} + \frac{\tan{\left(2 x \right)}}{x^{2}}\right)}{x}\right) = \frac{16}{3}$$
- 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
$$\left[100.53345156734, \infty\right)$$
Convexa en los intervalos
$$\left[-1.70345385344701, 1.70345385344701\right]$$