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$$2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \frac{\sin^{2}{\left(x \right)}}{x} - \frac{\cos^{2}{\left(x \right)}}{x} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{\cos^{2}{\left(x \right)}}{x^{3}}\right) = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 98.1748231018121$$
$$x_{2} = 74.6129159462597$$
$$x_{3} = -35.3425225656426$$
$$x_{4} = -40.0550060749993$$
$$x_{5} = -32.2008497455043$$
$$x_{6} = -11.7777902551183$$
$$x_{7} = -71.4711370117135$$
$$x_{8} = -49.4798861350585$$
$$x_{9} = 46.3387270601314$$
$$x_{10} = 62.0465858494662$$
$$x_{11} = 16.4953766419568$$
$$x_{12} = 886.714527112716$$
$$x_{13} = 76.1837097163166$$
$$x_{14} = -77.7543370485657$$
$$x_{15} = -79.3251355377163$$
$$x_{16} = 49.4802906228188$$
$$x_{17} = -98.1747188096946$$
$$x_{18} = -91.8915262229128$$
$$x_{19} = -38.484176676373$$
$$x_{20} = 35.3433351021862$$
$$x_{21} = 24.3482044996812$$
$$x_{22} = -57.3339151252179$$
$$x_{23} = 85.6084692436878$$
$$x_{24} = 8.64649825876728$$
$$x_{25} = 38.484861102782$$
$$x_{26} = -41.6258174775018$$
$$x_{27} = 88.7500570283684$$
$$x_{28} = 68.3297481010291$$
$$x_{29} = -24.3465494392149$$
$$x_{30} = -5.48498214421554$$
$$x_{31} = 40.0556219530264$$
$$x_{32} = 77.7545014175584$$
$$x_{33} = -55.7631130536459$$
$$x_{34} = 63.6173777375122$$
$$x_{35} = 33.7725660791654$$
$$x_{36} = -13.3490601683493$$
$$x_{37} = 47.9095127661744$$
$$x_{38} = 55.7634318632548$$
$$x_{39} = -68.3295354413945$$
$$x_{40} = 2.45918841915183$$
$$x_{41} = 25.9189284492171$$
$$x_{42} = -19.6336883606847$$
$$x_{43} = -82.4667340759196$$
$$x_{44} = -84.0375339339077$$
$$x_{45} = -25.9174088327353$$
$$x_{46} = 41.6264018693934$$
$$x_{47} = -33.7717015390395$$
$$x_{48} = 18.0657344963829$$
$$x_{49} = -10.2055879004712$$
$$x_{50} = -99.7455172442474$$
$$x_{51} = -76.1835362590277$$
$$x_{52} = -18.0627457306966$$
$$x_{53} = -46.3382661602626$$
$$x_{54} = 32.2018300513449$$
$$x_{55} = 82.4668820312663$$
$$x_{56} = -47.9090723548496$$
$$x_{57} = 54.1926483256621$$
$$x_{58} = -74.6127374903741$$
$$x_{59} = 99.7456172612756$$
$$x_{60} = 19.6363550807047$$
$$x_{61} = 30.6310702547146$$
$$x_{62} = 96.6040279552001$$
$$x_{63} = 5.51596461512633$$
$$x_{64} = -69.9003349318818$$
$$x_{65} = 91.8916453077865$$
$$x_{66} = 84.0376747079238$$
$$x_{67} = 79.3252954844258$$
$$x_{68} = -16.4915753784413$$
$$x_{69} = 44.767953342127$$
$$x_{70} = 10.2157438715929$$
$$x_{71} = 3.9740021610582$$
$$x_{72} = -63.6171286468689$$
$$x_{73} = 27.4896098475515$$
$$x_{74} = -60.4755229799467$$
$$x_{75} = -90.3207285066268$$
$$x_{76} = 52.6218592626052$$
$$x_{77} = 90.3208504243698$$
$$x_{78} = -54.1923045632751$$
$$x_{79} = 69.9005411014262$$
$$x_{80} = -3.89726154493934$$
$$x_{81} = -62.0463281195465$$
$$x_{82} = 60.4757987412468$$
$$x_{83} = -27.4883088267804$$
$$x_{84} = -2.30351813584042$$
$$x_{85} = -85.608331980016$$
$$x_{86} = 66.7589586368454$$
$$x_{87} = -93.4623251135572$$
$$x_{88} = 11.7847382599734$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \frac{\sin^{2}{\left(x \right)}}{x} - \frac{\cos^{2}{\left(x \right)}}{x} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{\cos^{2}{\left(x \right)}}{x^{3}}\right)\right) = -\infty$$
$$\lim_{x \to 0^+}\left(2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \frac{\sin^{2}{\left(x \right)}}{x} - \frac{\cos^{2}{\left(x \right)}}{x} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{\cos^{2}{\left(x \right)}}{x^{3}}\right)\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = 0$$
- 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:
Cóncava en los intervalos
$$\left[99.7456172612756, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -98.1747188096946\right]$$