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What is the value of Kc for critically stable system?
A question regarding the stability of my process

The characteristic equation of a closed loop system using a reverse acting proportional controller with gain Kc is:
12S^3 + 19S^2 + 8S + 1 + Kc = 0

What is the value of Kc which gives critically stable system?

The Kr that places the real part of the complex roots on the imaginary axis.

You can use Routh-Hurwitz or Root Locus methods.