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$$\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 63.6172512351933$$
$$x_{2} = -65.1880475619882$$
$$x_{3} = 47.9092879672443$$
$$x_{4} = 76.1836218495525$$
$$x_{5} = 57.3340659280137$$
$$x_{6} = 44.7676953136546$$
$$x_{7} = -11.7809724509617$$
$$x_{8} = -8.63937979737193$$
$$x_{9} = -62.0464549083984$$
$$x_{10} = -40.0553063332699$$
$$x_{11} = 3.92699081698724$$
$$x_{12} = 29.0597320457056$$
$$x_{13} = 38.484510006475$$
$$x_{14} = 13.3517687777566$$
$$x_{15} = -30.6305283725005$$
$$x_{16} = 32.2013246992954$$
$$x_{17} = 7.06858347057703$$
$$x_{18} = -43.1968989868597$$
$$x_{19} = -33.7721210260903$$
$$x_{20} = 88.7499924639117$$
$$x_{21} = -49.4800842940392$$
$$x_{22} = -58.9048622548086$$
$$x_{23} = -68.329640215578$$
$$x_{24} = 98.174770424681$$
$$x_{25} = -24.3473430653209$$
$$x_{26} = 35.3429173528852$$
$$x_{27} = 69.9004365423729$$
$$x_{28} = 85.6083998103219$$
$$x_{29} = -27.4889357189107$$
$$x_{30} = 54.1924732744239$$
$$x_{31} = -84.037603483527$$
$$x_{32} = 19.6349540849362$$
$$x_{33} = 10.2101761241668$$
$$x_{34} = 41.6261026600648$$
$$x_{35} = 79.3252145031423$$
$$x_{36} = -14.9225651045515$$
$$x_{37} = -46.3384916404494$$
$$x_{38} = -5.49778714378214$$
$$x_{39} = -96.6039740978861$$
$$x_{40} = 60.4756585816035$$
$$x_{41} = 51.0508806208341$$
$$x_{42} = 25.9181393921158$$
$$x_{43} = -90.3207887907066$$
$$x_{44} = -74.6128255227576$$
$$x_{45} = -55.7632696012188$$
$$x_{46} = 22.776546738526$$
$$x_{47} = 95.0331777710912$$
$$x_{48} = -93.4623814442964$$
$$x_{49} = -2.35619449019234$$
$$x_{50} = -71.4712328691678$$
$$x_{51} = -52.621676947629$$
$$x_{52} = -99.7455667514759$$
$$x_{53} = 66.7588438887831$$
$$x_{54} = -18.0641577581413$$
$$x_{55} = 0.785398163397448$$
$$x_{56} = 91.8915851175014$$
$$x_{57} = -36.9137136796801$$
$$x_{58} = -80.8960108299372$$
$$x_{59} = 101.316363078271$$
$$x_{60} = -87.1791961371168$$
$$x_{61} = 82.4668071567321$$
$$x_{62} = -21.2057504117311$$
$$x_{63} = 73.0420291959627$$
$$x_{64} = 16.4933614313464$$
$$x_{65} = -77.7544181763474$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -0.785398163397448$$
$$x_{2} = 2.35619449019234$$
$$\lim_{x \to -0.785398163397448^-}\left(\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = - \frac{1 \left(1.4142135623731 \sin{\left(0.785398163397448 + 0.25 \pi \right)} \cos{\left(0.785398163397448 + 0.25 \pi \right)} + 1.41421356237309 \sin^{2}{\left(0.785398163397448 + 0.25 \pi \right)}\right)}{\cos^{3}{\left(0.785398163397448 + 0.25 \pi \right)}}$$
$$\lim_{x \to -0.785398163397448^+}\left(\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = \frac{1 \left(- 1.4142135623731 \sin{\left(0.785398163397448 - 0.25 \pi \right)} \cos{\left(0.785398163397448 - 0.25 \pi \right)} + 1.41421356237309 \cos^{2}{\left(0.785398163397448 - 0.25 \pi \right)}\right)}{\sin^{3}{\left(0.785398163397448 - 0.25 \pi \right)}}$$
- los límites no son iguales, signo
$$x_{1} = -0.785398163397448$$
- es el punto de flexión
$$\lim_{x \to 2.35619449019234^-}\left(\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = \frac{1 \left(1.41421356237309 \sin{\left(0.25 \pi + 2.35619449019234 \right)} \cos{\left(0.25 \pi + 2.35619449019234 \right)} + 1.4142135623731 \cos^{2}{\left(0.25 \pi + 2.35619449019234 \right)}\right)}{\sin^{3}{\left(0.25 \pi + 2.35619449019234 \right)}}$$
$$\lim_{x \to 2.35619449019234^+}\left(\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = \frac{1 \left(1.41421356237309 \sin{\left(0.25 \pi + 2.35619449019234 \right)} \cos{\left(0.25 \pi + 2.35619449019234 \right)} + 1.4142135623731 \cos^{2}{\left(0.25 \pi + 2.35619449019234 \right)}\right)}{\sin^{3}{\left(0.25 \pi + 2.35619449019234 \right)}}$$
- 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
$$\left[101.316363078271, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -99.7455667514759\right]$$