Understanding the Concept:
A continuous control system crosses the imaginary axis when its closed-loop poles move from the left half of the s-plane onto the \(j\omega\) axis, signifying the boundary condition for marginal stability. This critical gain value can be found by evaluating the system's characteristic equation using the Routh-Hurwitz stability criterion. For a unity feedback configuration, the characteristic equation is:
\[
1 + G(s)H(s) = 0 \implies 1 + G(s) = 0
\]
Step 1: Construct the characteristic polynomial equation.
Given the open-loop transfer function:
\[
G(s) = \frac{K}{s(s+2)(s+4)}
\]
Setting up the closed-loop characteristic relation:
\[
1 + \frac{K}{s(s+2)(s+4)} = 0
\]
\[
s(s+2)(s+4) + K = 0
\]
Expanding the polynomial factor terms:
\[
s(s^2 + 6s + 8) + K = 0
\]
\[
s^3 + 6s^2 + 8s + K = 0
\]
Step 2: Construct the Routh Hurwitz array.
Using the coefficients of our third-order characteristic equation, we fill out the standard Routh rows:
{c|cc}
\(s^3\) & 1 & 8
\(s^2\) & 6 & \(K\)
\(s^1\) & \(\frac{(6 \times 8) - (1 \times K)}{6}\) & 0
\(s^0\) & \(K\) &
Simplifying the term in the \(s^1\) row:
\[
\frac{48 - K}{6}
\]
Step 3: Solve for marginal stability boundary condition.
For the system to cross the imaginary axis and sustain marginal oscillations, a complete row of zeros must appear in the array, or a coefficient in the primary column must equal zero.
Setting the \(s^1\) row element to zero:
\[
\frac{48 - K}{6} = 0
\]
\[
48 - K = 0 \implies K = 48
\]
Thus, when the amplifier loop gain \(K\) reaches exactly 48, the system crosses the imaginary axis. This matches option (D).