A manufacturer estimates the daily profit P (in hundreds of dollars) from producing q hundred units of a product according to
P(q)={−2q2+16q−15,10e−(q−4)+9,0≤q≤4,4<q≤9.
Assume the factory cannot produce fewer than 0 or more than 9 hundred units per day.
(a) Determine every critical value of P in the open interval (0,9). Show your analytical work.
(b) Using the candidates from part (a) together with all relevant endpoints or junction points, identify the production level(s) that yield the greatest daily profit. State both the production quantity and the corresponding maximum profit.
(c) Interpret your answer in part (b) for the production manager. Specify exactly how many individual units should be produced and the resulting profit in dollars.