Hello, Sales. Carmen Ramos, a colleague of the salesman has a erritory that can

Question

Hello,

Sales. Carmen Ramos, a colleague of the salesman has a erritory that can be described in terms of a rectangular grid as the region bounded by the curve y=x^2 and the line y = 16 , where x and y are in miles. She determines that the number of units S( x,y) she can sell at each grid point (x,y) in her region is given by the function

S(x,y)= (6x^2)-(36x)+(9y^2)-(6y) +60

At what piont(s) in Carmen's sales territory should she expect maximum sales to occur, and what are her maximum expected sales? Answer the same question for minimum sales.

Please show how you do it,

Thank you,

Explanation / Answer

Given y = x^2

Differentiatimg both sides wrt y we get
dy/dy = 2x*(dx/dy)

1 = 2x (dx/dy)

1/2x = dx/dy      ------------------------------------------(i)

S(x,y) = 6x^2 - 36x + 9y^2 - 6y + 60

dS(x,y)/dy = 12x dx/dy - 36dx/dy + 18y - 6

d^2S(x,y)/dy^2 = 12 d^2x/dy^2 - 36d^2x/dy^2 + 18

Putting d^2S(x,y)/dy^2 = 0

On solving we get (x,y) = (+1,-1); (1,1)

Max sales will happen at (x,y) = (-1,1) and Min sales will happen at (x,y) = (1,1)

At (x,y) = (1,1) she can expect maximum sales to occur

Min Sales = 6 (1)^2 - 36 (1) + 9 * (1)^2 - 6(1) + 60 = 33

Max Sales = 6 (-1)^2 - 36(-1) + 9*(1)^2 - 6(1) + 60 = 105

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