Chapter 4.6, Problem 39ES

Suppose that the position functions of two particles, $P_1$ and $P_2,$ in motion along the same line are $s_1=\frac{1}{2}{t}^2-t+3$ and $s_2=-\frac{1}{4}{t}^2+t+1$ respectively, for $t\ge 0$ .

(a) Prove that $P_1$ and $P_2$ do not collide.

(b) How close do $P_1$ and $P_2$ get to each other?

(c) During what intervals of time are they moving in opposite directions?