p(x) = (x-m)q
substitute to get your function
p(x) = 75+80(5m-x)(x-m)
dp/dx = 80(6m-2x)
to maximize equate to zero
80(6m-2x) = 0
x = 3m
3455 = 75 + 80(2m)(2m)
m = 3.25
A marketing researcher has determined that when a product sells at x dollars per unit' the number of units sold per week is given by the function
q= 75/x-m + 80(5m -x)
where m > 0 is the total manufactunng cost in dollar per unit and m < x <5m.
(a) Find the profit function P(x) (in dollars per week) and simplify it.
(b) Use calculus to find the selling price x that maximizes weekly profit. Sufficiently justify your solution
(c) Use your answer to part (b) to determine the manufacturing cost m that yields a maximum weekly profit of 3,455 dollar