could anyone help me with this?

Question

The total revenue for Fred's Estates LLC is given as the function R(x)=200x−0.25x^2, where x is the number of villas booked. What number of villas booked produces the maximum revenue?

Answer 1

R'(x) =200-.5x=0

x = 200/.5 = 400 villas produces maximum revenue

R(400) =200(400) -.25(400)^2

= 80000 -160000/4 = 80000-40000

=40000

R(x)=200x -.25x^2 = 0

.25x(800-x) =0

800-x=0 or .25x=0

x=800 or 0=

minimum revenue output

maximizing output level is the average (800+0)/2 = 400

rewrite the function in vertex form

-.25x^2 +200x =0

-.25(x^2 +800x +160000) + 40000 =0

-.25(x-400)^2 +40000 is the parabola in vertex form with

x=400=revenue maximizing output level

and 40,000= maximum revenue