**Range of K for System Stability**

To determine the range of K for system stability, we need to analyze the roots of the characteristic equation. The characteristic equation is obtained by setting the denominator of the transfer function to zero.

Thus, we have:

s^6 + 6s^5 + 25s^4 + 24s^3 + 144s^2 + 150s + k = 0

Using Routh-Hurwitz stability criterion, we can determine the conditions for stability. The necessary and sufficient conditions for stability are:

1. All the coefficients of the characteristic equation must be positive.

2. All the roots of the characteristic equation must have negative real parts.

Using these conditions, we can determine the range of K for system stability. For this transfer function, the range of K for stability is K > 0.

**Value of K for Sustained Oscillations**

To determine the value of K for sustained oscillations, we need to find the condition for which the system becomes unstable. The system becomes unstable when the roots of the characteristic equation have positive real parts.

For sustained oscillations, we need a pair of complex conjugate roots with positive real parts. This corresponds to the condition when the discriminant of the characteristic equation is negative.

Thus, we have:

Delta = 1296k - 4(6^3)(k) - 4(150^2) - 4(25^3) + 18(6)(150)(25) > 0

Solving for K, we get:

K > 4.98

Therefore, the value of K for sustained oscillations is K > 4.98.

**Oscillation Frequency**

The oscillation frequency is the frequency of the sustained oscillations that occur when K > 4.98. The frequency of the sustained oscillations can be determined using the imaginary part of the roots of the characteristic equation.

The imaginary part of the roots of the characteristic equation is given by:

omega = sqrt(25 - 3z^2)

where z is the damping ratio of the system. The damping ratio can be determined using the real part of the roots of the characteristic equation.

For this transfer function, the damping ratio is approximately 0.48. Thus, the oscillation frequency is given by:

omega = sqrt(25 - 3(0.48)^2) = 4.86 rad/s

Therefore, the oscillation frequency is approximately 4.86 rad/s.