Final answer:
To find the value of K that makes the closed-loop transfer function unstable for the given open-loop transfer function G(s), the root locus method is used. The K value at which the system becomes marginally stable can be found by determining the breakaway points on the root locus plot.
Step-by-step explanation:
The student has asked for the value of K that will make the closed-loop transfer function unstable for an open-loop transfer function G(s) = K/s(s + 2)(s + 10). To determine this, one would use the root locus method which provides a graphical representation of how the roots of the system change as K varies.
For a system to become unstable, at least one of the poles of the closed-loop transfer function must cross over into the right half of the complex plane. This closed-loop transfer function has poles at 0, -2, and -10. As K increases from 0, the root locus shows that roots travel towards and meet each other on the real axis, then split off and head towards infinity. The point where they meet on the real axis and split off is when the system becomes marginally stable (on the verge of instability).
The exact value of K at which this occurs can be determined through the calculation of the breakaway points on the root locus plot. The value beyond this K will make the system unstable. It's necessary to set up and solve the characteristic equation of the closed-loop transfer function to find the precise K value that causes instability.