Business Profit Analysis: A small business owner is analyzing their profit over time. Their profit function P (t) is given by

P (t) = 500t 0.02t^2, where t represents the number of days since the business started.

(a) Based on the background of the story, find the maximal domain of this function.

(b) Does P¹(t) exist? Why or why not?

(c) The function P(t) can be described as taking the time since the start of this business as input and outputting the profit of the business at that time. Find the derivative P(t). Explain what P(t) is measuring by giving a similar description.

(d) Find the time at which the business reaches its maximum profit and calculate the maximum profit.