For the cost and price functions below, find

a. the number, q, of units that produces maximum profit

b. the price, p, per unit that produces maximum profit

c. the maximum profit, P.

C(q) = 70 + 17q

p = 77 - 2q

a) The number, q, of units that produces maximum profit = 15

b) The price, p, per unit that produces maximum profit = 47 (currency not giben in the question)

c) Maximum Profit = P = 380 (currency not given in the question).

Explanation:

The cost function and price per unit function are given respectively as

C(q) = 70 + 17q

p = 77 - 2q

where q = quantity or number of units

a.) the number, q, of units that produces maximum profit

Total cost = C(q) = 70 + 17q

Revenue = (price per unit) × (Number of units) = p × q = (77 - 2q) × q = (77q - 2q²)

Profits = P(q) = (Revenue) - (Total Cost)

P(q) = (77q - 2q²) - (70 + 17q)

P(q) = -2q² + 60q - 70

To maximize the profits, we just obtain the point where the profit function reaches a Maximum.

At the maximum of a function, (dP/dq) = 0 and (d²P/dq²) < 0

Profit = P(q) = -2q² + 60q - 70

(dP/dq) = -4q + 60

At maximum point,

(dP/dq) = -4q + 60 = 0

q = (60/4) = 15

(d²P/dQ²) = -4 < 0 (hence, showing that the this point corresponds to a maximum point truly)

Hence, the number, q, of units that produces maximum profit = 15.

b.) the price, p, per unit that produces maximum profit

The price per unit is given as

p = 77 - 2q

Maximum profit occurs at q = 15

p = 77 - (2×15) = 47

Hence, the price, p, per unit that produces maximum profit = 47 (currency not given in the question)

c.) the maximum profit, P.

The Profit function is given as

Profit = P(q) = -2q² + 60q - 70

At maximum Profit, q = 15

Maximum Profit = P(15)

= -2(15²) + 60(15) - 70

= 380 (currency not given in the question).

Hope this Helps!!!

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