Finding the Maximum Magnitude Response (Gain) of Second-Order Filters without Calculus

The maximum gain (with respect to frequency) of second-order filters such as low-pass, high-pass, band-pass, low-pass notch and high-pass notch filters is derived without using calculus. Our method uses the fact that the square of the magnitude response (gain) for these filters can be written as a positive constant divided by a quadratic function of frequency. Hence, the gain is maximized when the denominator of the gain is minimized, which is easily achieved without calculus, as the denominator is parabolic.