Given information:
- Critical value of gain = 40
- Gain margin = 6 dB

To understand the answer, let's first define what gain and gain margin mean in the context of control systems.

Gain refers to the amplification of the input signal by the system. It is represented as a ratio of the output signal to the input signal. In this case, the gain is measured in decibels (dB).

Gain margin, on the other hand, is a measure of how much additional gain the system can tolerate before it becomes unstable. It is usually expressed in decibels (dB) and provides an indication of the stability of the system.

Now, let's analyze the given information in detail.

1. Critical value of gain = 40
The critical value of gain represents the gain at which the system becomes unstable. In other words, if the gain exceeds this value, the system will become unstable and may exhibit oscillations or even diverge.

2. Gain margin = 6 dB
The gain margin is a measure of how close the system is to the critical value of gain. A higher gain margin indicates a more stable system, as it means there is a larger buffer between the current gain and the critical value of gain.

Now, let's determine the operating gain of the system.

Since the gain margin is given as 6 dB, we can calculate the actual gain as follows:

Operating gain = Critical value of gain - Gain margin
= 40 - 6 dB
= 34 dB

To convert the gain from decibels to a linear scale, we use the formula:

Gain (linear scale) = 10^(Gain (dB)/20)

Therefore, the operating gain in linear scale is:

Operating gain (linear scale) = 10^(34/20)
= 20

Hence, the correct answer is option A) 20. The system is operating at a gain of 20.

In summary, the critical value of gain represents the gain at which the system becomes unstable. The gain margin tells us how close the system is to the critical value. By subtracting the gain margin from the critical value, we can determine the operating gain of the system. In this case, the operating gain is 20.