Final answer:
The maximum amplitude of a damped harmonic oscillator occurs at the driving frequency that equals the system's natural frequency, with the least damping resulting in the greatest response.
Step-by-step explanation:
The amplitude of a damped harmonic oscillator reaches its largest value when the driving frequency is equal to the natural frequency of the oscillator, a condition known as resonance. This is because at resonance, the energy supplied by the driving force is most effectively transferred to the oscillator, maximizing the amplitude. The least amount of damping in the system allows for the greatest response, reflected in the highest peak on the amplitude versus frequency graph. As damping increases, the peak amplitude decreases, and the resonance becomes broader. To find the driving frequency for the largest amplitude, one needs to adjust the driving frequency to match the natural frequency of the system.