Discussion
What happens to a synchronously detected AM signal if the local carrier has a small frequency error?
Answer: C
A frequency error makes the effective phase drift, so the recovered output is multiplied by a slow beat at the difference frequency, heard as a rise and fall or, for speech, as an unnatural warble. That sensitivity is the practical price of coherent detection and the reason a pilot tone or a phase-locked loop is used to derive the reference. The amplitude is certainly affected. Polarity inversion is what an exact half-cycle phase error would give. And the output does not vanish, except momentarily as the phase passes through quadrature.
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