Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d z^{2}} f{\left(z \right)} = 0$$
(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:
$$\frac{d^{2}}{d z^{2}} f{\left(z \right)} = $$
segunda derivada$$\frac{- \left(\frac{1}{z - 1} + \frac{1}{z}\right) \sin{\left(z \right)} + \cos{\left(z \right)} - \frac{\left(2 z - 1\right) \sin{\left(z \right)}}{z \left(z - 1\right)} + \frac{2 \cos{\left(z \right)}}{z \left(z - 1\right)} - \frac{2 \left(2 z - 1\right) \cos{\left(z \right)}}{z \left(z - 1\right)^{2}} - \frac{2 \left(2 z - 1\right) \cos{\left(z \right)}}{z^{2} \left(z - 1\right)}}{z \left(z - 1\right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$z_{1} = -73.7735634613355$$
$$z_{2} = 26.5497805529484$$
$$z_{3} = 64.3399786725715$$
$$z_{4} = 29.7080359002561$$
$$z_{5} = 10.5961128892338$$
$$z_{6} = -58.0511295101695$$
$$z_{7} = 95.7765886676923$$
$$z_{8} = 58.0499400526036$$
$$z_{9} = 54.9043251661795$$
$$z_{10} = -17.0501819888326$$
$$z_{11} = -70.6295891466443$$
$$z_{12} = -83.2044118692333$$
$$z_{13} = 42.3157932802559$$
$$z_{14} = -42.3180357527722$$
$$z_{15} = -54.9056551425221$$
$$z_{16} = 98.9195224021387$$
$$z_{17} = -13.8573762081368$$
$$z_{18} = 13.8357484426428$$
$$z_{19} = 89.4904367799601$$
$$z_{20} = -61.1962073592828$$
$$z_{21} = -3.74269333202927$$
$$z_{22} = -26.5555103582434$$
$$z_{23} = -664.440830648932$$
$$z_{24} = 83.203833494571$$
$$z_{25} = 70.6287861741731$$
$$z_{26} = 48.6115131429435$$
$$z_{27} = 3.22245492895961$$
$$z_{28} = -76.9173441131405$$
$$z_{29} = 51.7582153745922$$
$$z_{30} = 36.0156076750442$$
$$z_{31} = -98.9199314788638$$
$$z_{32} = 67.4845145811825$$
$$z_{33} = 17.0360783571991$$
$$z_{34} = -95.7770250534755$$
$$z_{35} = -39.1690153815412$$
$$z_{36} = -92.6340298908745$$
$$z_{37} = -29.7126034038903$$
$$z_{38} = -64.3409465615048$$
$$z_{39} = -7.33700145962957$$
$$z_{40} = 61.1951372459564$$
$$z_{41} = 45.4640934391097$$
$$z_{42} = -51.7597123769465$$
$$z_{43} = -86.3477347620627$$
$$z_{44} = -10.6338325169388$$
$$z_{45} = 20.2170137948893$$
$$z_{46} = -48.6132108060855$$
$$z_{47} = 73.7728275574363$$
$$z_{48} = -89.4909366807036$$
$$z_{49} = -183.761461365779$$
$$z_{50} = 92.6335633655438$$
$$z_{51} = -80.0609538176612$$
$$z_{52} = -45.466035099043$$
$$z_{53} = -36.0187078039572$$
$$z_{54} = -20.2269632150095$$
$$z_{55} = -32.8667450697016$$
$$z_{56} = 32.863017691483$$
$$z_{57} = 76.9166672063495$$
$$z_{58} = 7.25180774495468$$
$$z_{59} = 23.3868712872949$$
$$z_{60} = -636.166229610899$$
$$z_{61} = 86.3471977693559$$
$$z_{62} = 39.1663961017082$$
$$z_{63} = -23.394275987796$$
$$z_{64} = 80.0603290824966$$
$$z_{65} = -67.4853942378642$$
$$z_{66} = 177.477382508355$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$z_{1} = 0$$
$$z_{2} = 1$$
$$\lim_{z \to 0^-}\left(\frac{- \left(\frac{1}{z - 1} + \frac{1}{z}\right) \sin{\left(z \right)} + \cos{\left(z \right)} - \frac{\left(2 z - 1\right) \sin{\left(z \right)}}{z \left(z - 1\right)} + \frac{2 \cos{\left(z \right)}}{z \left(z - 1\right)} - \frac{2 \left(2 z - 1\right) \cos{\left(z \right)}}{z \left(z - 1\right)^{2}} - \frac{2 \left(2 z - 1\right) \cos{\left(z \right)}}{z^{2} \left(z - 1\right)}}{z \left(z - 1\right)}\right) = -\infty$$
$$\lim_{z \to 0^+}\left(\frac{- \left(\frac{1}{z - 1} + \frac{1}{z}\right) \sin{\left(z \right)} + \cos{\left(z \right)} - \frac{\left(2 z - 1\right) \sin{\left(z \right)}}{z \left(z - 1\right)} + \frac{2 \cos{\left(z \right)}}{z \left(z - 1\right)} - \frac{2 \left(2 z - 1\right) \cos{\left(z \right)}}{z \left(z - 1\right)^{2}} - \frac{2 \left(2 z - 1\right) \cos{\left(z \right)}}{z^{2} \left(z - 1\right)}}{z \left(z - 1\right)}\right) = \infty$$
- los límites no son iguales, signo
$$z_{1} = 0$$
- es el punto de flexión
$$\lim_{z \to 1^-}\left(\frac{- \left(\frac{1}{z - 1} + \frac{1}{z}\right) \sin{\left(z \right)} + \cos{\left(z \right)} - \frac{\left(2 z - 1\right) \sin{\left(z \right)}}{z \left(z - 1\right)} + \frac{2 \cos{\left(z \right)}}{z \left(z - 1\right)} - \frac{2 \left(2 z - 1\right) \cos{\left(z \right)}}{z \left(z - 1\right)^{2}} - \frac{2 \left(2 z - 1\right) \cos{\left(z \right)}}{z^{2} \left(z - 1\right)}}{z \left(z - 1\right)}\right) = \infty$$
$$\lim_{z \to 1^+}\left(\frac{- \left(\frac{1}{z - 1} + \frac{1}{z}\right) \sin{\left(z \right)} + \cos{\left(z \right)} - \frac{\left(2 z - 1\right) \sin{\left(z \right)}}{z \left(z - 1\right)} + \frac{2 \cos{\left(z \right)}}{z \left(z - 1\right)} - \frac{2 \left(2 z - 1\right) \cos{\left(z \right)}}{z \left(z - 1\right)^{2}} - \frac{2 \left(2 z - 1\right) \cos{\left(z \right)}}{z^{2} \left(z - 1\right)}}{z \left(z - 1\right)}\right) = -\infty$$
- los límites no son iguales, signo
$$z_{2} = 1$$
- 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
$$\left[98.9195224021387, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -636.166229610899\right]$$