Solution:
Given characteristic equation is,
6s^3 + 11s^2 + 6s + (1 - k) = 0

For the system to be stable, all the roots of the characteristic equation should lie in the left half of the s-plane.

To find the value of K beyond which the system just becomes unstable, we need to find the value of K for which at least one root of the characteristic equation lies in the right half of the s-plane.

Approach:
We will use Routh-Hurwitz criteria to find the range of K for which the system is stable.

Routh-Hurwitz Criteria:
Routh-Hurwitz criteria is used to determine the stability of a system by analyzing the location of the roots of the characteristic equation in the left half of the s-plane. Routh-Hurwitz criteria is based on the construction of a Routh array using the coefficients of the characteristic equation.

Steps to construct a Routh array:
Step 1: Write the coefficients of the characteristic equation in the first two rows of the Routh array.
Step 2: Construct the remaining rows of the Routh array by using the following formula:

R(i,j) = [R(i-1,1)R(i-2,j+1) - R(i-2,1)R(i-1,j+1)] / R(i-1,1)

where,
i = 3, 4, 5, … (number of rows in the Routh array)
j = 0, 1, 2, … (number of columns in the Routh array)
R(i,j) = element in the i-th row and j-th column of the Routh array
R(i-1,1) = element in the (i-1)-th row and first column of the Routh array
R(i-2,j+1) = element in the (i-2)-th row and (j+1)-th column of the Routh array
R(i-1,j+1) = element in the (i-1)-th row and (j+1)-th column of the Routh array

Step 3: Determine the range of K for which all the elements in the first column of the Routh array are positive.

If all the elements in the first column of the Routh array are positive, then all the roots of the characteristic equation lie in the left half of the s-plane and the system is stable.

If any of the elements in the first column of the Routh array are zero or negative, then the number of roots of the characteristic equation in the right half of the s-plane is equal to the number of sign changes in the first column of the Routh array. If this number is odd, then the system is unstable.

Let's apply the Routh-Hurwitz criteria to find the range of K for which the system is stable.

Construction of Routh Array:

Coefficients of the characteristic equation are,
a0 = 1 - k
a1 = 6
a2 = 11
a3 = 6

The Routh array for the given characteristic equation is,

| 1 - k | 6 |
| 6 | 11 |
| 3-k/6| 0 |

For the system to be stable,
All the elements in the first column of the Routh array should