Exercises 36 through 39 require the extreme value property.

Carmen Ramos, a colleague of the salesman in Example 7.3.7, has a territory that can be described in terms of a rectangular grid as the region bounded by the curve y = x² 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

At what point(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.

Data from Example 7.3.7.

Paul Johnson is a salesman whose territory borders on a lake and can be described in terms of a rectangular grid as the region bounded by the curve y = x² (the lakefront) and the lines y = 0 and x = 3 as shown in Figure 7.18, where x and y are in miles. He determines that the number of units S(x, y) he can sell at each grid point (x, y) in his region is given by the function S(x, y) = 4x² − 16x + 4y² − 4y + 20 At what point(s) in his sales territory should Paul expect maximum sales to occur, and what are his maximum expected sales? Answer the same question for minimum sales.

Figure 7.18

Transcribed Image Text:

S(x, y) = 6x² 36x +9y²-6y + 60