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$$\frac{\frac{\pi \left(\pi \left(x + 1\right) \left(\sin^{2}{\left(\frac{\pi x}{4} \right)} - \cos^{2}{\left(\frac{\pi x}{4} \right)}\right) - 8 \sin{\left(\frac{\pi x}{4} \right)} \cos{\left(\frac{\pi x}{4} \right)}\right)}{8} + \frac{\left(\pi \left(x + 1\right) \sin{\left(\frac{\pi x}{4} \right)} - 2 \cos{\left(\frac{\pi x}{4} \right)}\right) \cos{\left(\frac{\pi x}{4} \right)}}{x} + \frac{2 \left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x^{2}}}{x} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 44.9996028704428$$
$$x_{2} = -31.0008536093723$$
$$x_{3} = -85.0001143749339$$
$$x_{4} = -15.0036931994676$$
$$x_{5} = -23.0015572519595$$
$$x_{6} = 18.9979377413296$$
$$x_{7} = -13.0054432541212$$
$$x_{8} = 96.9999141708959$$
$$x_{9} = 10.9942059886459$$
$$x_{10} = 52.9997133756277$$
$$x_{11} = 38.9994888660358$$
$$x_{12} = -51.000313896626$$
$$x_{13} = -37.0006189745222$$
$$x_{14} = -57.0002567704716$$
$$x_{15} = 74.9998590011155$$
$$x_{16} = 26.9989529792397$$
$$x_{17} = -35.0006687223095$$
$$x_{18} = -25.0013851152113$$
$$x_{19} = -65.0001967550027$$
$$x_{20} = 80.9998770033365$$
$$x_{21} = 88.999898081113$$
$$x_{22} = 40.9995219605996$$
$$x_{23} = -3.0944985600091$$
$$x_{24} = -27.0011272880282$$
$$x_{25} = -89.0001042345009$$
$$x_{26} = -91.0000982779299$$
$$x_{27} = -67.0001815606124$$
$$x_{28} = 72.9998486406245$$
$$x_{29} = -59.0002343114102$$
$$x_{30} = -39.0005379923011$$
$$x_{31} = -43.0004421616485$$
$$x_{32} = -47.0003698293801$$
$$x_{33} = -1.614150703963$$
$$x_{34} = 102.999924798238$$
$$x_{35} = -69.000174348217$$
$$x_{36} = 92.9999066437242$$
$$x_{37} = 50.9996981653844$$
$$x_{38} = 90.9999038579305$$
$$x_{39} = -61.0002237758992$$
$$x_{40} = 2.94416537967508$$
$$x_{41} = 380.999994421377$$
$$x_{42} = 70.9998428586052$$
$$x_{43} = 16.9972505899113$$
$$x_{44} = -87.0001075427893$$
$$x_{45} = -11.0069285099769$$
$$x_{46} = 104.999926730846$$
$$x_{47} = 32.9992635820045$$
$$x_{48} = 58.9997734936612$$
$$x_{49} = -93.0000953851793$$
$$x_{50} = -9.01200286161397$$
$$x_{51} = 98.9999186505986$$
$$x_{52} = 12.9953233745264$$
$$x_{53} = 14.9967639467234$$
$$x_{54} = -21.0019878533421$$
$$x_{55} = -95.0000901607308$$
$$x_{56} = -83.0001181823166$$
$$x_{57} = 48.9996648501043$$
$$x_{58} = 94.9999117171202$$
$$x_{59} = -45.0004151584059$$
$$x_{60} = 86.9998949005943$$
$$x_{61} = -49.0003490998428$$
$$x_{62} = -19.0022898196266$$
$$x_{63} = -5.04433678704151$$
$$x_{64} = 62.9998009955008$$
$$x_{65} = 76.9998639235117$$
$$x_{66} = -63.000205420042$$
$$x_{67} = 66.9998237765818$$
$$x_{68} = -17.0030898885942$$
$$x_{69} = -53.0002976385393$$
$$x_{70} = -55.0002697559914$$
$$x_{71} = 68.999830630629$$
$$x_{72} = 60.999783437987$$
$$x_{73} = 56.9997520770949$$
$$x_{74} = 78.9998727782642$$
$$x_{75} = 84.9998882840567$$
$$x_{76} = -79.0001304825544$$
$$x_{77} = -71.0001616289714$$
$$x_{78} = 82.999884630574$$
$$x_{79} = 64.9998092038565$$
$$x_{80} = 30.999199612962$$
$$x_{81} = 54.99973987063$$
$$x_{82} = -41.0005019053572$$
$$x_{83} = -75.0001448080093$$
$$x_{84} = 36.9994135457222$$
$$x_{85} = -29.0010200422274$$
$$x_{86} = 42.9995779111432$$
$$x_{87} = -73.00015556184$$
$$x_{88} = 100.99992082309$$
$$x_{89} = -7.01740170165913$$
$$x_{90} = -81.0001260704169$$
$$x_{91} = 46.999645563784$$
$$x_{92} = 28.9990477660696$$
$$x_{93} = 34.9993683584672$$
$$x_{94} = 8.99032715593967$$
$$x_{95} = -77.0001396559479$$
$$x_{96} = -33.0007823359871$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\frac{\pi \left(\pi \left(x + 1\right) \left(\sin^{2}{\left(\frac{\pi x}{4} \right)} - \cos^{2}{\left(\frac{\pi x}{4} \right)}\right) - 8 \sin{\left(\frac{\pi x}{4} \right)} \cos{\left(\frac{\pi x}{4} \right)}\right)}{8} + \frac{\left(\pi \left(x + 1\right) \sin{\left(\frac{\pi x}{4} \right)} - 2 \cos{\left(\frac{\pi x}{4} \right)}\right) \cos{\left(\frac{\pi x}{4} \right)}}{x} + \frac{2 \left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x^{2}}}{x}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{\frac{\pi \left(\pi \left(x + 1\right) \left(\sin^{2}{\left(\frac{\pi x}{4} \right)} - \cos^{2}{\left(\frac{\pi x}{4} \right)}\right) - 8 \sin{\left(\frac{\pi x}{4} \right)} \cos{\left(\frac{\pi x}{4} \right)}\right)}{8} + \frac{\left(\pi \left(x + 1\right) \sin{\left(\frac{\pi x}{4} \right)} - 2 \cos{\left(\frac{\pi x}{4} \right)}\right) \cos{\left(\frac{\pi x}{4} \right)}}{x} + \frac{2 \left(x + 1\right) \cos^{2}{\left(\frac{\pi x}{4} \right)}}{x^{2}}}{x}\right) = \infty$$
- 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
$$\left[380.999994421377, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -95.0000901607308\right]$$