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$$3 \left(3 x^{2} \left(x - 4\right)^{2} + \left(x - 2\right) \left(x^{3} - 6 x^{2} + 24\right)\right) \left(\frac{x^{3}}{36} - \frac{x^{2}}{6} + \frac{2}{3}\right) = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 2 - \frac{\sqrt[3]{108 + 108 \sqrt{3} i}}{3} - \frac{12}{\sqrt[3]{108 + 108 \sqrt{3} i}}$$
$$x_{2} = 2 - \frac{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}{2} - \frac{\sqrt{12 - 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}} + \frac{4}{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}} - \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}}{2}$$
$$x_{3} = 2 - \frac{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}{2} + \frac{\sqrt{12 - 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}} + \frac{4}{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}} - \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}}{2}$$
$$x_{4} = 2 - \frac{\sqrt{12 - 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}} - \frac{4}{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}} - \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}}{2} + \frac{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}{2}$$
$$x_{5} = 2 + \frac{\sqrt{12 - 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}} - \frac{4}{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}} - \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}}{2} + \frac{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}{2}$$
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
$$\left(-\infty, 2 - \frac{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}{2} - \frac{\sqrt{12 - 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}} + \frac{4}{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}} - \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}}{2}\right] \cap \left(-\infty, 2 + \frac{\sqrt{12 - 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}} - \frac{4}{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}} - \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}}{2} + \frac{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}{2}\right] \cap \left[2 - 4 \cos{\left(\frac{\pi}{9} \right)}, \infty\right) \cap \left[2 - \frac{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}{2} + \frac{\sqrt{12 - 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}} + \frac{4}{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}} - \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}}{2}, \infty\right) \cap \left[2 - \frac{\sqrt{12 - 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}} - \frac{4}{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}} - \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}}{2} + \frac{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}{2}, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, 2 - 4 \cos{\left(\frac{\pi}{9} \right)}\right] \cap \left(-\infty, 2 - \frac{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}{2} + \frac{\sqrt{12 - 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}} + \frac{4}{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}} - \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}}{2}\right] \cap \left(-\infty, 2 - \frac{\sqrt{12 - 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}} - \frac{4}{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}} - \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}}{2} + \frac{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}{2}\right] \cap \left[2 - \frac{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}{2} - \frac{\sqrt{12 - 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}} + \frac{4}{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}} - \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}}{2}, \infty\right) \cap \left[2 + \frac{\sqrt{12 - 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}} - \frac{4}{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}} - \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}}{2} + \frac{\sqrt{6 + \frac{25}{2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}} + 2 \sqrt[3]{\frac{119}{8} + \frac{\sqrt{366} i}{4}}}}{2}, \infty\right)$$