(1) Apple is considering manufacturing a new high-power laptop. They estimate that the revenue earned from selling their laptop at p dollars per laptop would be R(p) = 200v p, where R is given in billions of dollars per year. They also estimate that the cost incurred by supplying for demand at price p is given by C(p) = 2p + 100, where C is also given in billions of dollars per year. (a) Find the yearly profit function P(p) in terms of p, the price per laptop. (b)Use the First Derivative Test to identify the relative and absolute extrema (if any) of P(p). part (b) continued. (c)At what price should Apple sell their laptop to maximize their yearly profit? (d)What is the resulting maximum yearly revenue? (2) For this problem, assume that R(x) and C(x) represent the revenue and cost associated to selling or making x units of a good. (a)Using the general equation P(x) = R(x) - C(x), demonstrates that P' (a) = 0 exactly when R' (a) = C'(a). (b) Explain in a sentence or two why your conclusion in part (a) makes sense (hint: as we discussed in class, think about P' (a), R' (a), and C'(a) as the approximated profit/revenue/cost of the a + 1st unit)