find the maximum revenue.

Question

The revenue function for a bicycle shop is given by R(x)=x•p(x) dollars where x is the number of units sold and p(x)=500 -0.5x is the unit price. find the maximum revenue. 

Answer 1

R(x) = xp(x) = x(500 - 0.5x)

 

                        = -0.5x² + 500x

 

The graph of R(x) is a parabola opening downward (since the coefficient of x² is negative.

 

If A is negative, then the maximum of y = Ax² + Bx + C occurs                                        when x = -B/(2A). 

 

So, the maximum of R(x) occurs at

  x = -500/[2(-0.5)] = -500/ (-1) = 500

 

  Maximum revenue = R(500) = -0.5(500)² + 500(500)

 

                                                     = $125,000