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(A) For resonance,the amplitude $a_m$ must be maximum. This occurs when the denominator $f(\omega) = a\omega^2 - b\omega + c$ is minimum.To find the m

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$b^2 = 4ac$

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(A) For resonance,the amplitude $a_m$ must be maximum. This occurs when the denominator $f(\omega) = a\omega^2 - b\omega + c$ is minimum.To find the minimum,we set the derivative with respect to $\omega$ to zero:$df/d\omega = 2a\omega - b = 0 \Rightarrow \omega = b/(2a)$.However,for a single resonant frequency to exist in the physical domain,the quadratic expression $a\omega^2 - b\omega + c$ must have a single real root or the minimum value must be zero at the resonance frequency.Mathematically,for the denominator to reach zero (leading to infinite amplitude in an ideal case) or to have a unique stationary point that defines the resonance,the discriminant of the quadratic equation $a\omega^2 - b\omega + c = 0$ must be zero.Thus,$D = b^2 - 4ac = 0$,which implies $b^2 = 4ac$.

The amplitude of vibration of a particle is given by $a_m = a_0 / (a\omega^2 - b\omega + c)$,where $a_0$,$a$,$b$,and $c$ are positive constants. The condition for a single resonant frequency is:

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