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$$\frac{4 \left(\frac{4 x^{2} \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}}{x^{2} - 4} + \frac{2 x^{2}}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{2}} - \frac{4 x^{2} \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{3}} - \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}\right)}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -2192.80038673447$$
$$x_{2} = 973.011918966819$$
$$x_{3} = 1280.13853339472$$
$$x_{4} = -2499.55769570708$$
$$x_{5} = -1169.8640783696$$
$$x_{6} = -348.290332826221$$
$$x_{7} = 1177.79062797554$$
$$x_{8} = 2405.22644862719$$
$$x_{9} = 665.446434146895$$
$$x_{10} = 562.70381723382$$
$$x_{11} = 1024.21953910942$$
$$x_{12} = -296.252212973538$$
$$x_{13} = 408.101000198785$$
$$x_{14} = -2397.3083554515$$
$$x_{15} = -606.14064426858$$
$$x_{16} = -760.107795397665$$
$$x_{17} = -503.279883774574$$
$$x_{18} = -1425.70225506308$$
$$x_{19} = -1886.00766643811$$
$$x_{20} = -2346.18260014419$$
$$x_{21} = 459.729184681486$$
$$x_{22} = -1579.15892212912$$
$$x_{23} = 2251.84714076265$$
$$x_{24} = 921.793283620639$$
$$x_{25} = 1587.08047328897$$
$$x_{26} = 2047.32855755532$$
$$x_{27} = -811.376338135048$$
$$x_{28} = -1272.21369170668$$
$$x_{29} = 819.314745520888$$
$$x_{30} = 1996.19614962672$$
$$x_{31} = 870.561697985203$$
$$x_{32} = 1433.62518807934$$
$$x_{33} = 2558.59915083526$$
$$x_{34} = 511.254137212811$$
$$x_{35} = 1484.77995583004$$
$$x_{36} = -2243.92867210584$$
$$x_{37} = 251.986947164547$$
$$x_{38} = -1323.38077376337$$
$$x_{39} = -2448.4333722096$$
$$x_{40} = -2295.05605696655$$
$$x_{41} = 2302.97439209109$$
$$x_{42} = -400.093655348202$$
$$x_{43} = -1118.68013767305$$
$$x_{44} = 1075.41771189355$$
$$x_{45} = 2098.45977994229$$
$$x_{46} = -1016.28940352763$$
$$x_{47} = -1732.59215523488$$
$$x_{48} = 1740.51267391462$$
$$x_{49} = -1528.00965161165$$
$$x_{50} = 1638.22678526966$$
$$x_{51} = -862.625914310343$$
$$x_{52} = -1681.44995253779$$
$$x_{53} = -913.859698631662$$
$$x_{54} = -1783.73240253261$$
$$x_{55} = 2149.58990131392$$
$$x_{56} = -965.080194719043$$
$$x_{57} = -2090.54085039906$$
$$x_{58} = 1945.06246275068$$
$$x_{59} = 356.325819136067$$
$$x_{60} = 304.331063214959$$
$$x_{61} = 199.036243956251$$
$$x_{62} = -2550.68136760381$$
$$x_{63} = -1937.14295890515$$
$$x_{64} = 1842.79082675193$$
$$x_{65} = 1791.65263460783$$
$$x_{66} = 716.761640626397$$
$$x_{67} = -657.496198025057$$
$$x_{68} = 2354.1008103501$$
$$x_{69} = -1630.30561067982$$
$$x_{70} = -2141.67113641196$$
$$x_{71} = -451.741132162319$$
$$x_{72} = 768.049368568156$$
$$x_{73} = 2507.47557595137$$
$$x_{74} = 1331.3049054257$$
$$x_{75} = 614.096923275279$$
$$x_{76} = -554.739773254891$$
$$x_{77} = -190.752975112801$$
$$x_{78} = -1834.87085773074$$
$$x_{79} = 1689.37078427002$$
$$x_{80} = -243.836163251503$$
$$x_{81} = 1382.46697024754$$
$$x_{82} = 1893.92739344449$$
$$x_{83} = -1067.48894347188$$
$$x_{84} = -1988.27685199681$$
$$x_{85} = 2200.71899834771$$
$$x_{86} = 1228.96731747796$$
$$x_{87} = 2456.35135559302$$
$$x_{88} = -708.816199295763$$
$$x_{89} = 1126.6077210965$$
$$x_{90} = -1374.54347118903$$
$$x_{91} = 1535.93161762305$$
$$x_{92} = -1476.85753131522$$
$$x_{93} = -2039.40945086897$$
$$x_{94} = -1221.0416751731$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -2$$
$$x_{2} = 2$$
$$\lim_{x \to -2^-}\left(\frac{4 \left(\frac{4 x^{2} \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}}{x^{2} - 4} + \frac{2 x^{2}}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{2}} - \frac{4 x^{2} \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{3}} - \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}\right)}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{2}}\right) = 32 - 2 \pi$$
$$\lim_{x \to -2^+}\left(\frac{4 \left(\frac{4 x^{2} \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}}{x^{2} - 4} + \frac{2 x^{2}}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{2}} - \frac{4 x^{2} \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{3}} - \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}\right)}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{2}}\right) = 2 \pi + 32$$
- los límites no son iguales, signo
$$x_{1} = -2$$
- es el punto de flexión
$$\lim_{x \to 2^-}\left(\frac{4 \left(\frac{4 x^{2} \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}}{x^{2} - 4} + \frac{2 x^{2}}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{2}} - \frac{4 x^{2} \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{3}} - \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}\right)}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{2}}\right) = 2 \pi + 32$$
$$\lim_{x \to 2^+}\left(\frac{4 \left(\frac{4 x^{2} \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}}{x^{2} - 4} + \frac{2 x^{2}}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{2}} - \frac{4 x^{2} \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{3}} - \operatorname{atan}{\left(\frac{1}{x^{2} - 4} \right)}\right)}{\left(1 + \frac{1}{\left(x^{2} - 4\right)^{2}}\right) \left(x^{2} - 4\right)^{2}}\right) = 32 - 2 \pi$$
- los límites no son iguales, signo
$$x_{2} = 2$$
- es el punto de flexión
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