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$$\frac{- \left(\frac{\log{\left(x \right)} + 2}{x \log{\left(x \right)} + x - 3} + \frac{1}{x}\right) \left(x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3\right) - \frac{\left(\log{\left(x \right)} + 2\right) \left(x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3\right)}{x \log{\left(x \right)} + x - 3} + 2 \log{\left(x \right)} + 5 - \frac{x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3}{x}}{x^{2} \left(x \log{\left(x \right)} + x - 3\right)^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 3083.01279622447$$
$$x_{2} = 5357.17083543138$$
$$x_{3} = 1395.14427660911$$
$$x_{4} = 3649.56462473852$$
$$x_{5} = 5547.50664569847$$
$$x_{6} = 8604.8129104997$$
$$x_{7} = 8413.17431392798$$
$$x_{8} = 7838.65369462865$$
$$x_{9} = 6882.55047288283$$
$$x_{10} = 4217.56921314887$$
$$x_{11} = 6691.56392415366$$
$$x_{12} = 1209.01064533135$$
$$x_{13} = 6500.66115746419$$
$$x_{14} = 2142.74527775831$$
$$x_{15} = 2330.33986079912$$
$$x_{16} = 5166.94219284043$$
$$x_{17} = 7455.98750923113$$
$$x_{18} = 4407.19047333876$$
$$x_{19} = 1023.18115505367$$
$$x_{20} = 8796.51457367029$$
$$x_{21} = 4976.82502762394$$
$$x_{22} = 3271.68715367179$$
$$x_{23} = 3460.54100103902$$
$$x_{24} = 1955.41141596735$$
$$x_{25} = 8221.60042105109$$
$$x_{26} = 1768.3555947259$$
$$x_{27} = 8988.27774009301$$
$$x_{28} = 4028.08649965968$$
$$x_{29} = 1581.5949680715$$
$$x_{30} = 2518.17863873065$$
$$x_{31} = 7264.76465782928$$
$$x_{32} = 4596.94399285195$$
$$x_{33} = 2706.2462488441$$
$$x_{34} = 4786.82396767143$$
$$x_{35} = 7647.28455238836$$
$$x_{36} = 9180.10091548322$$
$$x_{37} = 7073.61820345915$$
$$x_{38} = 6119.11815517186$$
$$x_{39} = 9371.98267031054$$
$$x_{40} = 6309.84492717354$$
$$x_{41} = 3838.74915993612$$
$$x_{42} = 8030.09294782234$$
$$x_{43} = 2894.52859015622$$
$$x_{44} = 5737.94560228732$$
$$x_{45} = 5928.48394576449$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$x_{2} = 1.85455071909613$$
$$\lim_{x \to 0^-}\left(\frac{- \left(\frac{\log{\left(x \right)} + 2}{x \log{\left(x \right)} + x - 3} + \frac{1}{x}\right) \left(x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3\right) - \frac{\left(\log{\left(x \right)} + 2\right) \left(x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3\right)}{x \log{\left(x \right)} + x - 3} + 2 \log{\left(x \right)} + 5 - \frac{x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3}{x}}{x^{2} \left(x \log{\left(x \right)} + x - 3\right)^{2}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{- \left(\frac{\log{\left(x \right)} + 2}{x \log{\left(x \right)} + x - 3} + \frac{1}{x}\right) \left(x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3\right) - \frac{\left(\log{\left(x \right)} + 2\right) \left(x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3\right)}{x \log{\left(x \right)} + x - 3} + 2 \log{\left(x \right)} + 5 - \frac{x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3}{x}}{x^{2} \left(x \log{\left(x \right)} + x - 3\right)^{2}}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = 0$$
- es el punto de flexión
$$\lim_{x \to 1.85455071909613^-}\left(\frac{- \left(\frac{\log{\left(x \right)} + 2}{x \log{\left(x \right)} + x - 3} + \frac{1}{x}\right) \left(x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3\right) - \frac{\left(\log{\left(x \right)} + 2\right) \left(x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3\right)}{x \log{\left(x \right)} + x - 3} + 2 \log{\left(x \right)} + 5 - \frac{x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3}{x}}{x^{2} \left(x \log{\left(x \right)} + x - 3\right)^{2}}\right) = 5.14445688441069 \cdot 10^{47}$$
$$\lim_{x \to 1.85455071909613^+}\left(\frac{- \left(\frac{\log{\left(x \right)} + 2}{x \log{\left(x \right)} + x - 3} + \frac{1}{x}\right) \left(x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3\right) - \frac{\left(\log{\left(x \right)} + 2\right) \left(x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3\right)}{x \log{\left(x \right)} + x - 3} + 2 \log{\left(x \right)} + 5 - \frac{x \left(\log{\left(x \right)} + 2\right) + x \log{\left(x \right)} + x - 3}{x}}{x^{2} \left(x \log{\left(x \right)} + x - 3\right)^{2}}\right) = 5.14445688441069 \cdot 10^{47}$$
- 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:
No tiene corvaduras en todo el eje numérico