Find the maximum revenue.

Question

The demand function for a new product is p(x)=-7x+65, where p is the prices in dollars and x is the number of items sold in thousands. The cost function is c(x)=9x+105. Determine the maximum revenue.

Answer 1

I think you are actually asking for the maximum profit.

Revenue: R(x) = xp(x) = -7x²+65x

Profit: P(x) = R(x)-C(x) = -7x²+56x-105

This is a parabola of the form ax²+bx+C

The vertex of the parabola is the maximum (or minimum) of the expression. This can be found by

x = -b/(2a) = -56/(2(-7)) = 4

P(4) = $7

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If you want the maximum revenue without factoring in the cost:

x = -65/(2(-7)) = 4.64....

R(4.64) ≈ $151