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$$\frac{512 \left(\tan^{2}{\left(32 x \right)} + 1\right) \tan{\left(32 x \right)} - \frac{16 \left(\tan^{2}{\left(32 x \right)} + 1\right)}{x} + \frac{\tan{\left(32 x \right)}}{2 x^{2}}}{x} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -69.9986252639607$$
$$x_{2} = 16.0025486043484$$
$$x_{3} = 42.0188249828175$$
$$x_{4} = -17.7696884032881$$
$$x_{5} = -41.9206502668213$$
$$x_{6} = 53.9961418193494$$
$$x_{7} = -55.7632871138538$$
$$x_{8} = 64.0099655733006$$
$$x_{9} = 60.2793252413693$$
$$x_{10} = -64.0099655733006$$
$$x_{11} = -67.7406060092314$$
$$x_{12} = 8.24679913056223$$
$$x_{13} = -91.9897705039147$$
$$x_{14} = 56.2541608131621$$
$$x_{15} = 28.0780191217217$$
$$x_{16} = 46.2403379892888$$
$$x_{17} = 21.9911929819905$$
$$x_{18} = -23.7583355466728$$
$$x_{19} = -39.7608065829084$$
$$x_{20} = 78.245304509249$$
$$x_{21} = 40.2516801355094$$
$$x_{22} = 86.0011102472499$$
$$x_{23} = -80.0124501012444$$
$$x_{24} = 24.2492085667399$$
$$x_{25} = 66.1698100246605$$
$$x_{26} = 94.2477899693411$$
$$x_{27} = -11.6828812689226$$
$$x_{28} = 80.0124501012444$$
$$x_{29} = 79.7179258350631$$
$$x_{30} = 26.0163516989573$$
$$x_{31} = -71.7657707880746$$
$$x_{32} = 58.0213061520816$$
$$x_{33} = -51.7381228889027$$
$$x_{34} = -48.0074830795384$$
$$x_{35} = -32.0050056712123$$
$$x_{36} = -93.7569161714671$$
$$x_{37} = 34.2630233801181$$
$$x_{38} = 74.2201395987041$$
$$x_{39} = -57.9231314101842$$
$$x_{40} = 32.0050056712123$$
$$x_{41} = -59.9848010096399$$
$$x_{42} = -75.9872851603569$$
$$x_{43} = 2.25845201290999$$
$$x_{44} = 90.1244500855665$$
$$x_{45} = 44.3750182389745$$
$$x_{46} = 100.236450346187$$
$$x_{47} = 69.9986252639607$$
$$x_{48} = -77.7544307359183$$
$$x_{49} = 6.08699619491471$$
$$x_{50} = 88.2591296765058$$
$$x_{51} = -49.8728029567908$$
$$x_{52} = -3.82907106292848$$
$$x_{53} = 72.2566445477549$$
$$x_{54} = -10.0139241025127$$
$$x_{55} = 37.9936618576287$$
$$x_{56} = 20.1258764596351$$
$$x_{57} = -81.7795957051525$$
$$x_{58} = 98.2729551323501$$
$$x_{59} = 18.2605607781088$$
$$x_{60} = -37.9936618576287$$
$$x_{61} = -29.7469882675837$$
$$x_{62} = -86.0011102472499$$
$$x_{63} = -7.7559327725012$$
$$x_{64} = -19.7331783435486$$
$$x_{65} = -16.0025486043484$$
$$x_{66} = -27.9798444733323$$
$$x_{67} = -95.7204113663014$$
$$x_{68} = 68.2314797576412$$
$$x_{69} = 14.235410312076$$
$$x_{70} = -83.7430908336606$$
$$x_{71} = 12.2719258793527$$
$$x_{72} = -45.7494643637664$$
$$x_{73} = 48.0074830795384$$
$$x_{74} = -89.8299258098194$$
$$x_{75} = -99.7455765420091$$
$$x_{76} = -43.9823193537665$$
$$x_{77} = 10.0139241025127$$
$$x_{78} = -5.98882405413538$$
$$x_{79} = -33.7721499422649$$
$$x_{80} = 91.9897705039147$$
$$x_{81} = 52.2289965636289$$
$$x_{82} = -61.7519464113978$$
$$x_{83} = -21.9911929819905$$
$$x_{84} = 4.12357716381613$$
$$x_{85} = -97.7820813301124$$
$$x_{86} = 36.2265172437986$$
$$x_{87} = 96.0149356462803$$
$$x_{88} = 75.9872851603569$$
$$x_{89} = -87.7682558862655$$
$$x_{90} = 84.2339646178271$$
$$x_{91} = 62.2428201388031$$
$$x_{92} = -13.7445389099122$$
$$x_{93} = -53.9961418193494$$
$$x_{94} = -1.76769808630633$$
$$x_{95} = -65.8752857793704$$
$$x_{96} = -72.9438678133976$$
$$x_{97} = 30.237861586773$$
$$x_{98} = -25.820002443554$$
$$x_{99} = -35.7356437619636$$
$$x_{100} = 50.2655018855128$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{512 \left(\tan^{2}{\left(32 x \right)} + 1\right) \tan{\left(32 x \right)} - \frac{16 \left(\tan^{2}{\left(32 x \right)} + 1\right)}{x} + \frac{\tan{\left(32 x \right)}}{2 x^{2}}}{x}\right) = \frac{16384}{3}$$
$$\lim_{x \to 0^+}\left(\frac{512 \left(\tan^{2}{\left(32 x \right)} + 1\right) \tan{\left(32 x \right)} - \frac{16 \left(\tan^{2}{\left(32 x \right)} + 1\right)}{x} + \frac{\tan{\left(32 x \right)}}{2 x^{2}}}{x}\right) = \frac{16384}{3}$$
- 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
$$\left[100.236450346187, \infty\right)$$
Convexa en los intervalos
$$\left[-1.76769808630633, 2.25845201290999\right]$$