Envelope Detector:
- The envelope detector is a method used to detect the envelope of an amplitude-modulated (AM) signal.
- It is based on the principle that the envelope of the modulated signal is proportional to the magnitude of the modulating signal.
- In this case, the modulated signal g(t) is given as g(t) = (3 to 6 cos 2pfmt) cos 2pfct, where fmt is the modulating frequency and fct is the carrier frequency.
- The envelope of this signal is the magnitude of the modulating signal, which is given by the expression |3 to 6 cos 2pfmt|.
- The envelope detector can be implemented using a diode and a low-pass filter.
- The diode rectifies the modulated signal, converting it into a positive signal.
- The low-pass filter removes the high-frequency carrier component, leaving only the envelope signal.
- The output of the envelope detector is the detected envelope signal, which can then be further processed or demodulated.

Synchronous Detector:
- The synchronous detector, also known as the coherent detector, is a method used to demodulate a modulated signal.
- It requires synchronization with the carrier signal in order to recover the modulating signal.
- In this case, the modulated signal g(t) is given as g(t) = (3 to 6 cos 2pfmt) cos 2pfct, where fct is the carrier frequency and fct is the modulating frequency.
- The synchronous detector multiplies the modulated signal with a synchronized local oscillator signal, which is at the same frequency as the carrier signal.
- The multiplication process results in the sum and difference frequencies.
- By using a low-pass filter, the sum frequency component (2fct + fmt) is removed, leaving only the difference frequency component (2fct - fmt).
- The output of the synchronous detector is the demodulated signal, which is the difference frequency component.

Ratio Detector:
- The ratio detector is a method used to demodulate a frequency-modulated (FM) signal.
- It is not suitable for demodulating an amplitude-modulated (AM) signal like the one given in this question.
- Therefore, the ratio detector is not the most suitable method for detecting the modulated signal g(t) = (3 to 6 cos 2pfmt) cos 2pfct.

Conclusion:
- The most suitable method for detecting the modulated signal g(t) = (3 to 6 cos 2pfmt) cos 2pfct is the envelope detector.
- The envelope detector is based on the principle that the envelope of the modulated signal is proportional to the magnitude of the modulating signal.
- It can be implemented using a diode and a low-pass filter to extract the envelope signal.
- The output of the envelope detector is the detected envelope signal, which can then be further processed or demodulated.