A manufacturer is planning to sell a new product at the pric

A manufacturer is planning to sell a new product at the price of $230 per unit and estimates that if x thousand dollars is spent on development and y thousand dollars is spent on promotion, consumers will buy approximately ((150y)/(y+4))+((300x)/(x+2)) units of the product. If manufacturing costs $180 per unit, how much should the manufacturer spend on development and how much on promotion to generate the largest possible profit from the sale of this product?

Promotion (y):

Development (x):

Solution

profit z=((150y)/(y+4))+((300x)/(x+2))*(230-180)-(1000x+1000y) to maximize z dz/dx=0 dz/dy=0 so we get x=sqrt(30)-2,y=sqrt(30)-4 dev=3477.23 promotion:1477.23