The number of times a new pop song has been downloaded $t$ weeks after its initial release is given by $f(t)=\frac{4,000,000}{1+300 e^{-0.55 t}}$
a) After how many weeks is the rate of change of the number of downloads maximized?
b) What is the rate of change of the number of downloads at the time found in part (a)?
c) How many times has the song been downloaded at the time found in part (a)?
a) The rate of change of the number of downloads is maximized after $\square$ weeks. (Do not round until the final answer. Then round to two decimal places as needed.)

Step 1 ：The rate of change of the number of downloads is given by the derivative of the function \(f(t)\). To find the maximum rate of change, we need to find the critical points of the derivative, which are the points where the derivative is either 0 or undefined. We can do this by setting the derivative equal to 0 and solving for \(t\).

Step 2 ：The derivative of the function is a rational function, and rational functions are undefined when the denominator is equal to 0. So, we need to find the values of \(t\) for which the denominator of the derivative, \((1 + 300 e^{-0.55 t})^2\), is equal to 0.

Step 3 ：The values of \(t\) for which the denominator of the derivative is equal to 0 are complex numbers, which do not make sense in the context of this problem. Therefore, the derivative of the function is never undefined.

Step 4 ：This means that the maximum rate of change occurs at one of the critical points of the derivative. However, we found earlier that the derivative has no real roots, which means that the derivative is always positive.

Step 5 ：Therefore, the rate of change of the number of downloads is always increasing, and the maximum rate of change occurs as \(t\) approaches infinity.

Step 6 ：Final Answer: The rate of change of the number of downloads is maximized after \(\boxed{\infty}\) weeks.