A mobile robot suitable for nighttime guard duty is available. This guard never sleeps and can tirelessly patrol large warehouses and outdoor yards. The steering control system for the mobile robot has a unity feedback with the loop transfer function
$$
G_c(s) G(s)=\frac{K(s+1)(s+5)}{s(s+1.5)(s+2)} .
$$
(a) Find $K$ for all breakaway and entry points on the real axis. (b) Find $K$ when the damping ratio of the complex roots is 0.707 . (c) Find the minimum value of the damping ratio for the complex roots and the associated gain $K$. (d) Find the overshoot and the time to settle (to within $2 \%$ of the final value) for a unit step input for the gain, $K$, determined in parts (b) and (c).