[Math] Let $r$, $s$, and $t$ be the roots of the equation $x^3 – 2x + 1 = 0$ in some order. What is the maximal value of $r^3 – s- t$

Question

Let $r$, $s$, and $t$ be the roots of the equation $x^3 – 2x + 1 = 0$ in some order. What is the maximal value of $r^3 – s- t$?

How should I approach this problem? I have no idea how to start, any answer is greatly appreciated.

Answer 1

for any or the three roots.

$r^3 - 2r + 1 = 0\\ r^3 = 2r - 1$

substitute: $2r - s - t - 1$

$r + s + t = 0$ from Vieta.

Substitute again

$3r - 1$

What is the largest of the 3 roots?