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$$\frac{10 \left(- \frac{60 x^{2} \left(5 x^{2} + 1\right)}{\left(5 x^{2} + 1\right)^{2} + 1} - \frac{20 x^{2}}{\left(\left(5 x^{2} + 1\right)^{2} + 1\right) \operatorname{atan}{\left(5 x^{2} + 1 \right)}} + 3\right)}{9 \left(\left(5 x^{2} + 1\right)^{2} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(5 x^{2} + 1 \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 5468.98928314651$$
$$x_{2} = -1728.78232109488$$
$$x_{3} = 9176.04223659447$$
$$x_{4} = -10014.536466638$$
$$x_{5} = -2164.69980356109$$
$$x_{6} = -857.633092165753$$
$$x_{7} = -5435.22218524723$$
$$x_{8} = -7833.88720402694$$
$$x_{9} = -9142.27372486186$$
$$x_{10} = 5905.10397206832$$
$$x_{11} = -1946.72854640712$$
$$x_{12} = -4344.95916315284$$
$$x_{13} = -5653.27887028152$$
$$x_{14} = 3070.48943956995$$
$$x_{15} = 2634.45195838359$$
$$x_{16} = -8051.95083728184$$
$$x_{17} = 3724.59090882189$$
$$x_{18} = -9796.4704776597$$
$$x_{19} = -4126.91214954942$$
$$x_{20} = -2600.69218041638$$
$$x_{21} = -1293.01587035651$$
$$x_{22} = 10920.5708619083$$
$$x_{23} = 9612.17326247065$$
$$x_{24} = 4160.67764571156$$
$$x_{25} = 9830.23908937137$$
$$x_{26} = 5032.8790620255$$
$$x_{27} = 891.316193383403$$
$$x_{28} = 8085.71912519377$$
$$x_{29} = -8706.14372019004$$
$$x_{30} = 4378.72502907637$$
$$x_{31} = -2818.70557328527$$
$$x_{32} = 2416.44717410563$$
$$x_{33} = -3472.7886143245$$
$$x_{34} = -5871.33656176024$$
$$x_{35} = -7615.82394862551$$
$$x_{36} = -7179.69870913049$$
$$x_{37} = 8957.97706819408$$
$$x_{38} = 10702.5041915001$$
$$x_{39} = -3908.86772778437$$
$$x_{40} = 8739.91215241213$$
$$x_{41} = -7397.76110449904$$
$$x_{42} = -3036.72713706388$$
$$x_{43} = 3288.51843092318$$
$$x_{44} = -8488.07912084226$$
$$x_{45} = 6123.16269353565$$
$$x_{46} = 9394.10764001912$$
$$x_{47} = 2198.45539207976$$
$$x_{48} = -6307.45454637495$$
$$x_{49} = -6961.63680470053$$
$$x_{50} = 1762.5301887478$$
$$x_{51} = 8521.84750864398$$
$$x_{52} = 4814.82600643865$$
$$x_{53} = -9578.40468182239$$
$$x_{54} = -4563.00839679567$$
$$x_{55} = -3254.75522763638$$
$$x_{56} = 3506.55255472256$$
$$x_{57} = -5217.16663290873$$
$$x_{58} = -10450.66897565$$
$$x_{59} = 7649.59211903637$$
$$x_{60} = 8303.78315832266$$
$$x_{61} = 7431.52920823206$$
$$x_{62} = -2382.68920560774$$
$$x_{63} = -6089.39515152365$$
$$x_{64} = 6995.40475583469$$
$$x_{65} = -9360.33909262677$$
$$x_{66} = -3690.82635760031$$
$$x_{67} = 1544.61330170871$$
$$x_{68} = 7213.46674003171$$
$$x_{69} = 5250.93354450875$$
$$x_{70} = 10048.3051074092$$
$$x_{71} = 1108.96139070981$$
$$x_{72} = 1980.48091658493$$
$$x_{73} = -1510.87200805012$$
$$x_{74} = -10668.7354739717$$
$$x_{75} = 0.292570661043346$$
$$x_{76} = -1075.24710682186$$
$$x_{77} = 4596.77458090568$$
$$x_{78} = -4999.11236155708$$
$$x_{79} = 3942.63279082879$$
$$x_{80} = -4781.05954655229$$
$$x_{81} = -6525.51466557913$$
$$x_{82} = 5687.04613339481$$
$$x_{83} = 10266.3713044031$$
$$x_{84} = -10232.6026364077$$
$$x_{85} = 6777.34330236989$$
$$x_{86} = 2852.46675902649$$
$$x_{87} = -6743.57543884741$$
$$x_{88} = 10484.4376691858$$
$$x_{89} = -8270.01481849455$$
$$x_{90} = 7867.65543563596$$
$$x_{91} = 6559.28243258186$$
$$x_{92} = 6341.2222066999$$
$$x_{93} = 1326.74701967758$$
$$x_{94} = -10886.8021218124$$
$$x_{95} = -8924.20859476241$$
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
$$\left(-\infty, 0.292570661043346\right]$$
Convexa en los intervalos
$$\left[0.292570661043346, \infty\right)$$