Finding Marginal Revenue, Marginal Cost, and Marginal Profit with Cost Function.

Question

A gizmo is sold for $28 per item. Suppose that the number of items produced is equal to the number of items sold and that the cost (in dollars) of producing 𝑥 gizmos is given by the following function: C(x)=6x^3+2x^2+9x+3

Explain and demonstrate how to find the marginal revenue, marginal cost, and marginal profit in this situation.

Answer 1

C(x) = 6x^3 +2x^2 +9x + 3

take the derivative

MC = C'(x) = 18x^2 +4x + 9 = Marginal Cost

R(x) = 28x

take the derivative

MR = 28 = Marginal Revenue

MP = MR-MC = 18x^2 +4x + 9 -28 = 18x^2 +4x -19 = Marginal Profit

Profit = R-C = -6x^3 -2x^2 +19x +3

MP = R' -C' = 19 - 18x^2 -4x = 0

18x^2 +4x -19 = 0

x = -4/36 + (1/36)sqr(16+4(18)(19)) = about 0.92 or x=1 for profit maximizing output

P(0) = $3 loss

P(1) = -6-2-+19 +3 = -8 = $14 profit

P(2) = -48-8+38+3 = $15 loss

the more produced the greater the loss without limit

only way to avoid bankruptcy is produce 1 gizmo