To find the value of K in the given equation, we need to solve the characteristic equation s^3 + Ks^2 + 5s + 10 = 0 for critical stability.

1. Characteristic Equation:
The characteristic equation of a feedback system is obtained by substituting s with the Laplace transform variable in the closed-loop transfer function of the system. In this case, the characteristic equation is given as:
s^3 + Ks^2 + 5s + 10 = 0

2. Critical Stability:
A system is said to be critically stable if all the roots of the characteristic equation have real parts equal to or less than zero. In other words, the system is stable, and any small perturbation in the system settles over time.

3. Finding the Value of K:
To determine the value of K for critical stability, we need to analyze the characteristic equation and check the stability conditions.

Let's consider K = 1:
s^3 + s^2 + 5s + 10 = 0

By using the Routh-Hurwitz stability criterion, we can analyze the stability of the system. The Routh-Hurwitz criteria state that for a system to be stable, all the coefficients of the characteristic equation must be positive.

Applying the Routh-Hurwitz criteria, we construct the Routh array:

1 5
1 10

From the first row of the Routh array, we can see that the coefficients are positive. However, in the second row, the second element is negative (10). Therefore, the system is not stable for K = 1.

Now, let's consider K = 2:
s^3 + 2s^2 + 5s + 10 = 0

Constructing the Routh array:

1 5
2 10

From the Routh array, we can observe that all the coefficients are positive. Hence, the system is stable for K = 2.

Therefore, the correct answer is option 'B' (K = 2) for critical stability.