The demand equation for a certain product is x^3 +4p=500, where x represents qua

Question

The demand equation for a certain product is x^3 +4p=500, where x represents quantity and p represents price. A) find the revenue function, R(x). B) find the quantity, x, that yields the maximum revenue. C) explain how you know that your answer to b gives a maximum. The demand equation for a certain product is x^3 +4p=500, where x represents quantity and p represents price. A) find the revenue function, R(x). B) find the quantity, x, that yields the maximum revenue. C) explain how you know that your answer to b gives a maximum. A) find the revenue function, R(x). B) find the quantity, x, that yields the maximum revenue. C) explain how you know that your answer to b gives a maximum.

Explanation / Answer

x^3 + 4p = 500

p = (500 - x^3) / 4

For x quantity, revenue, R(x) = x * p

a) R(x) = (500x - x^4) / 4

b)

R'(x) = (500 - 4x^3) / 4 = 0

500 - 4x^3 = 0

x^3 = 125

x = 5

c)

R''(x) = second derivative = -12x^2/4 --> -3x^2

When x = 5 : second derivative = negative value

So, we know that part B represented MAXIMUM

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