Question 168
On the Cartesian plane, let $C$ be the graph of the function $y = \sqrt[3]{x^2}$. A line $d$ varies on the plane such that $d$ always cuts $C$ at three distinct points with $x$-coordinates $x_1, x_2$ and $x_3$.
a. Prove that the following value is a constant:
$$
\sqrt[3]{\frac{x_1 x_2}{x_3^2}} + \sqrt[3]{\frac{x_2 x_3}{x_1^2}} + \sqrt[3]{\frac{x_3 x_1}{x_2^2}}
$$
b. Show that:
$$
\sqrt[3]{\frac{x_1^2}{x_2 x_3}} + \sqrt[3]{\frac{x_2^2}{x_3 x_1}} + \sqrt[3]{\frac{x_3^2}{x_1 x_2}} \lt -\frac{15}{4}
$$
a. Prove that the following value is a constant:
$$
\sqrt[3]{\frac{x_1 x_2}{x_3^2}} + \sqrt[3]{\frac{x_2 x_3}{x_1^2}} + \sqrt[3]{\frac{x_3 x_1}{x_2^2}}
$$
b. Show that:
$$
\sqrt[3]{\frac{x_1^2}{x_2 x_3}} + \sqrt[3]{\frac{x_2^2}{x_3 x_1}} + \sqrt[3]{\frac{x_3^2}{x_1 x_2}} \lt -\frac{15}{4}
$$