Question 4
In the Cartesian plane $Oxy$, let $(C)$ be the graph of $y=\sqrt[3]{x^2}$. A variable line $d$ meets $(C)$ at three distinct points with abscissas $x_1,x_2,x_3$.
(a) Prove that $\sqrt[3]{\dfrac{x_1x_2}{x_3^2}}+\sqrt[3]{\dfrac{x_2x_3}{x_1^2}}+\sqrt[3]{\dfrac{x_3x_1}{x_2^2}}$ is constant.
(b) Prove that $\sqrt[3]{\dfrac{x_1^2}{x_2x_3}}+\sqrt[3]{\dfrac{x_2^2}{x_3x_1}}+\sqrt[3]{\dfrac{x_3^2}{x_1x_2}}\lt -\dfrac{15}4$.
(a) Prove that $\sqrt[3]{\dfrac{x_1x_2}{x_3^2}}+\sqrt[3]{\dfrac{x_2x_3}{x_1^2}}+\sqrt[3]{\dfrac{x_3x_1}{x_2^2}}$ is constant.
(b) Prove that $\sqrt[3]{\dfrac{x_1^2}{x_2x_3}}+\sqrt[3]{\dfrac{x_2^2}{x_3x_1}}+\sqrt[3]{\dfrac{x_3^2}{x_1x_2}}\lt -\dfrac{15}4$.