?(a) Show that if the profit ????(????) is a maximum, then the marginal revenue equals

Chapter 4, Problem 66

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QUESTION:

(a) Show that if the profit ๐‘ƒ(๐‘ฅ) is a maximum, then the marginal revenue equals the marginal cost.

(b) If ๐ถ(๐‘ฅ) ๏ผ 16,000 + 500๐‘ฅ โˆ’ 1.6x2ย + 0.004๐‘ฅ3ย is the cost function and ๐‘(๐‘ฅ) ๏ผ 1700 โˆ’ 7๐‘ฅ is the demand function, find the production level that will maximize profit.

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QUESTION:

(a) Show that if the profit ๐‘ƒ(๐‘ฅ) is a maximum, then the marginal revenue equals the marginal cost.

(b) If ๐ถ(๐‘ฅ) ๏ผ 16,000 + 500๐‘ฅ โˆ’ 1.6x2ย + 0.004๐‘ฅ3ย is the cost function and ๐‘(๐‘ฅ) ๏ผ 1700 โˆ’ 7๐‘ฅ is the demand function, find the production level that will maximize profit.

ANSWER:

Step 1 of 2

a)

By definition, profit is the difference between revenue and cost, i.e.

If the profit ย is a maximum, then

Therefore if profit is maximum, then

By definition, ย is marginal revenue and ย is marginal cost.

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