Final answer:
The system is stable with a gain (K) of 0 as all poles are in the left-half of the s-plane. The system remains stable for all positive values of K since adjusting K will not cause the poles to cross into the right-half of the s-plane.
Step-by-step explanation:
To produce the Root Locus of the given transfer function G(s) = K(s+3)(s+5)/(s+1)(s+7), we need to look at the open-loop poles and zeros and how they move in the complex plane as K varies from 0 to infinity.
(a) When K=0, the transfer function G(s) reduces to zero. In this case, the stability of the system only depends on the poles of the transfer function, which are at s = -1 and s = -7. As all the poles are in the left-half of the s-plane, the system is stable at K=0.
(b) To determine if different values of K affect stability, we can examine how the root locus enters into the right-half of the s-plane. For the system to become unstable, the root locus must cross over the imaginary axis into the right-half plane. By adjusting K, we can vary the position of the closed-loop poles. However, since all the poles and zeros are already on the left side, increasing K will just move the poles along the real axis towards the zeros, but it will not cause the system to become unstable.