Question 53
For each real number $t$, let $P_t(x) = x^3 - 12x + t$ and let
$$
\Delta(t) = \max\{c \in \mathbb{R} \mid P_t(c) = 0\} - \min\{c \in \mathbb{R} \mid P_t(c) = 0\}
$$
the difference between the largest and the smallest real roots of $P_t(x)$. Determine the range of values that $\Delta(t)$ can assume as $t$ varies.
$$
\Delta(t) = \max\{c \in \mathbb{R} \mid P_t(c) = 0\} - \min\{c \in \mathbb{R} \mid P_t(c) = 0\}
$$
the difference between the largest and the smallest real roots of $P_t(x)$. Determine the range of values that $\Delta(t)$ can assume as $t$ varies.