The demand function for a product is p = 1000 - 2q Finding the level of production that maximizes total revenue producer, and determine that income
where p is the price (in
dollars) per unit when q units are
demand (per week) by
Take note that Revenue function (R(x)) is price times demand.
So, R(x) = (1000 - 2q)q = 1000q - 2q^2.
That is a quadratic function, which is in the form aq^2 + bq + c
where a < 0. Hence, the parabola opens downward. So, the maximum value will be found at the vertex.
So, we will solve for q, on vertex (q, r).
So, q = -b/2a = -1000/2(-2) = 250.
Hence, for 250 units per week, there will be maximum revenue.
And for the maximum revenue, we will replace the q by 250 on our revenue function.
R(250) = 1000(250) - 2(250)^2 = 250000 - 125000 = $125000.