Question

A company finds that the rate at which the quantity of a product that consumers demand changes with respect to price is given by the​ marginal-demand function Upper D prime (x )equals negative StartFraction 4000 Over x squared EndFraction where x is the price per​ unit, in dollars. Find the demand function if it is known that 1002 units of the product are demanded by consumers when the price is ​$4 per unit.

Answer 1

Answer: the demand function is D(x) = 4000/x + 2

Step-by-step explanation:

Given that

D'(x) = - 4000 / x²

Now d(D(x)/dx = - 4000/x²

so we integrate with respect to x

∫ d(D(x)) = - ∫ (4000/x²) dx = -4000 x⁻¹/-1 + C

⇒ D(x) = 4000/x + C

where C is a constant

Given that; x = 4 and D(x) = 1002

so we substitute

1002 = 4000 / 4 + C

1002 = 1000 + C

C = 1002 - 1000

C = 2

so  D(x) = 4000/x + C ⇒ D(x) = 4000/x + 2

Therefore the demand function is D(x) = 4000/x + 2