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$$\frac{2 \left(\frac{x - 1}{x + 1} - 1\right) \left(\frac{2 \left(x - 1\right) \left(\frac{x - 1}{x + 1} - 1\right)}{\left(x + 1\right) \left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right)} + \frac{3 \left(\frac{x - 1}{x + 1} - 1\right)}{\left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right) \operatorname{atan}{\left(\frac{x - 1}{x + 1} \right)}} - 2\right)}{\left(x + 1\right)^{2} \left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right) \operatorname{atan}^{3}{\left(\frac{x - 1}{x + 1} \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -27029.4148971184$$
$$x_{2} = -22793.716930171$$
$$x_{3} = -24487.8944087724$$
$$x_{4} = 25545.2257625422$$
$$x_{5} = -21946.6915457938$$
$$x_{6} = 28934.3644686674$$
$$x_{7} = -19405.9398392386$$
$$x_{8} = 23003.6065373258$$
$$x_{9} = -38892.1521836074$$
$$x_{10} = 18768.2493646861$$
$$x_{11} = 41645.4351882857$$
$$x_{12} = 23850.7860472359$$
$$x_{13} = 24697.9933352965$$
$$x_{14} = -32113.0946919035$$
$$x_{15} = 32323.7615312024$$
$$x_{16} = 22156.4578273115$$
$$x_{17} = 35713.3458095488$$
$$x_{18} = 19615.2340912683$$
$$x_{19} = 34018.5336453148$$
$$x_{20} = 20462.2672568572$$
$$x_{21} = -43129.3552142019$$
$$x_{22} = -25335.0375823493$$
$$x_{23} = -42281.9010884526$$
$$x_{24} = 29781.6928241211$$
$$x_{25} = 39103.0700866526$$
$$x_{26} = 27239.757050609$$
$$x_{27} = -21099.7156155475$$
$$x_{28} = 39950.5189782089$$
$$x_{29} = 33171.1422118906$$
$$x_{30} = -36349.9251948329$$
$$x_{31} = -28723.8947896457$$
$$x_{32} = 21309.3433937365$$
$$x_{33} = -34655.1567403154$$
$$x_{34} = 40797.9741359718$$
$$x_{35} = -30418.4593733517$$
$$x_{36} = 37408.1927518611$$
$$x_{37} = -29571.1674457004$$
$$x_{38} = -26182.2121095319$$
$$x_{39} = 30629.0358615931$$
$$x_{40} = -31265.7689398699$$
$$x_{41} = -23640.7861505459$$
$$x_{42} = -20252.7957605551$$
$$x_{43} = -39739.578376954$$
$$x_{44} = 34865.935073885$$
$$x_{45} = -41434.4532947861$$
$$x_{46} = 31476.3924393896$$
$$x_{47} = 36560.7652259908$$
$$x_{48} = 38255.627864874$$
$$x_{49} = -32960.4353315037$$
$$x_{50} = -38044.7341926607$$
$$x_{51} = -35502.5355234316$$
$$x_{52} = 28087.0520689961$$
$$x_{53} = 50083.4516590066$$
$$x_{54} = -37197.3249843191$$
$$x_{55} = -33807.7896958931$$
$$x_{56} = -27876.6432458536$$
$$x_{57} = 26392.4810140345$$
$$x_{58} = -40587.0122425022$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$\lim_{x \to -1^-}\left(\frac{2 \left(\frac{x - 1}{x + 1} - 1\right) \left(\frac{2 \left(x - 1\right) \left(\frac{x - 1}{x + 1} - 1\right)}{\left(x + 1\right) \left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right)} + \frac{3 \left(\frac{x - 1}{x + 1} - 1\right)}{\left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right) \operatorname{atan}{\left(\frac{x - 1}{x + 1} \right)}} - 2\right)}{\left(x + 1\right)^{2} \left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right) \operatorname{atan}^{3}{\left(\frac{x - 1}{x + 1} \right)}}\right) = - \frac{1 \left(-24 + 8 \pi\right)}{\pi^{4}}$$
$$\lim_{x \to -1^+}\left(\frac{2 \left(\frac{x - 1}{x + 1} - 1\right) \left(\frac{2 \left(x - 1\right) \left(\frac{x - 1}{x + 1} - 1\right)}{\left(x + 1\right) \left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right)} + \frac{3 \left(\frac{x - 1}{x + 1} - 1\right)}{\left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right) \operatorname{atan}{\left(\frac{x - 1}{x + 1} \right)}} - 2\right)}{\left(x + 1\right)^{2} \left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right) \operatorname{atan}^{3}{\left(\frac{x - 1}{x + 1} \right)}}\right) = \frac{1 \left(24 + 8 \pi\right)}{\pi^{4}}$$
- los límites no son iguales, signo
$$x_{1} = -1$$
- es el punto de flexión
$$\lim_{x \to 1^-}\left(\frac{2 \left(\frac{x - 1}{x + 1} - 1\right) \left(\frac{2 \left(x - 1\right) \left(\frac{x - 1}{x + 1} - 1\right)}{\left(x + 1\right) \left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right)} + \frac{3 \left(\frac{x - 1}{x + 1} - 1\right)}{\left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right) \operatorname{atan}{\left(\frac{x - 1}{x + 1} \right)}} - 2\right)}{\left(x + 1\right)^{2} \left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right) \operatorname{atan}^{3}{\left(\frac{x - 1}{x + 1} \right)}}\right) = \infty$$
$$\lim_{x \to 1^+}\left(\frac{2 \left(\frac{x - 1}{x + 1} - 1\right) \left(\frac{2 \left(x - 1\right) \left(\frac{x - 1}{x + 1} - 1\right)}{\left(x + 1\right) \left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right)} + \frac{3 \left(\frac{x - 1}{x + 1} - 1\right)}{\left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right) \operatorname{atan}{\left(\frac{x - 1}{x + 1} \right)}} - 2\right)}{\left(x + 1\right)^{2} \left(\frac{\left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} + 1\right) \operatorname{atan}^{3}{\left(\frac{x - 1}{x + 1} \right)}}\right) = \infty$$
- 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:
No tiene corvaduras en todo el eje numérico