Exercises 36 through 39 require the extreme value property.

In this exercise, you are asked to re-examine Exercise 29 as an optimization problem over a closed, bounded region.

a. In Exercise 29, explain why the variables x and y must satisfy

Show that the set of all points (x, y) that satisfy these inequalities is a triangular region, and find its vertices.

b. Solve Exercise 29 by finding the largest value of the profit function

over the triangular region found in part (a). Does this change the solution found in Exercise 29? For what values of x and y is the smallest profit obtained?

Data from Exercises 29

In Exercises 29 through 35, assume that the required extreme value is a relative extremum.

A T-shirt shop carries two competing shirts, one endorsed by Tim Duncan and the other by LeBron James. The owner of the store can obtain both types at a cost of $2 per shirt and estimates that if Duncan shirts are sold for x dollars apiece and James shirts for y dollars apiece, consumers will buy 40 − 50x + 40y Duncan shirts and 20 + 60x − 70y James shirts each day. How should the owner price the shirts to generate the largest possible profit?

Transcribed Image Text:

x ≥ 2, y = 2 40 50x + 40y = 0 20+ 60x70y = 0