1. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Maximum-Minimum (Optimization) Problems OBJECTIVE  Solve maximum and minimum problems using calculus. 4.5

2. A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose? How do we know that there will be a max at this value and not a min?

3. A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

4. A Procedure for Solving Applied Maximum and Minimum Problems (p. 276)

5. Example 2: What dimensions for a one liter cylindrical can will use the least amount of material? We can minimize the material by minimizing the area. area of ends lateral area We need another equation that relates r and h :

6. Example 2: What dimensions for a one liter cylindrical can will use the least amount of material? area of ends lateral area Is this min or max?

7. If the end points or places where the derivative fails to exist could be the maximum or minimum, you have to check. Notes: If the function that you want to optimize has more than one variable, use substitution to rewrite the function. Always check to see if the extreme you’ve found is a maximum or a minimum. You may check using the second derivative test. 

8. Slide 2.5 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3: From a thin piece of cardboard 8 in. by 8 in., square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume? 1 st make a drawing in which x is the length of each square to be cut.

9. Slide 2.5 - 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 (continued): 2 nd write an equation for the volume of the box. Note that x must be between 0 and 4. So, we need to maximize the volume equation on the interval (0, 4).

10. Slide 2.5 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Thus, the volume is maximized when the square corners are inches. The maximum volume is

11. Find the area of the largest rectangle that can be inscribed in a semicircle of radius r. Optimization Problems – Ex. 4

12. Let’s take the semicircle to be the upper half of the circle x 2 + y 2 = r 2 with center the origin. Then, the word inscribed means that the rectangle has two vertices on the semicircle and two vertices on the x-axis. Optimization Problems – Ex. 4

13. Let (x, y) be the vertex that lies in the first quadrant. Then, the rectangle has sides of lengths 2x and y. So, its area is: A = 2xy Optimization Problems – Ex. 4

14. To eliminate y, we use the fact that (x, y) lies on the circle x 2 + y 2 = r 2. So, Thus, Optimization Problems – Ex. 4

15. The domain of this function is 0 ≤ x ≤ r. Its derivative is: This is 0 when 2x 2 = r 2, that is x =, (since x ≥ 0). Optimization Problems – Ex. 4

16. This value of x gives a maximum value of A, since A(0) = 0 and A(r) = 0. Thus, the area of the largest inscribed rectangle is: Optimization Problems – Ex. 4

17. Slide 2.5 - 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5: A stereo manufacturer determines that in order to sell x units of a new stereo, the price per unit, in dollars, must be The manufacturer also determines that the total cost of producing x units is given by C(x) = 3000 + 20x a) Find the total revenue R(x). b) Find the total profit P(x). c) How many units must the company produce and sell in order to maximize profit? d) What is the maximum profit? e) What price per unit must be charged in order to make this maximum profit?

18. Slide 2.5 - 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley a) b)

19. Slide 2.5 - 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley c) Since there is only one critical value, we can use the second derivative to determine whether or not it yields a maximum or minimum. Since P  (x) is negative, x = 490 yields a maximum. Thus, profit is maximized when 490 units are bought and sold.

20. Slide 2.5 - 20 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5 (concluded): d) The maximum profit is given by Thus, the stereo manufacturer makes a maximum profit of $237,100 when 490 units are bought and sold. e) The price per unit to achieve this maximum profit is

21. Slide 2.5 - 21 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6: Promoters of international fund-raising concerts must walk a fine line between profit and loss, especially when determining the price to charge for admission to closed-circuit TV showings in local theaters. By keeping records, a theater determines that, at an admission price of $26, it averages 1000 people in attendance. For every drop in price of $1, it gains 50 customers. Each customer spends an average of $4 on concessions. What admission price should the theater charge in order to maximize total revenue?

22. Slide 2.5 - 22 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Let x = the number of dollars by which the price of $26 should be decreased (if x is negative, the price should be increased).

23. Slide 2.5 - 23 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To maximize R(x), we find R (x) and solve for critical values. Since there is only one critical value, we can use the second derivative to determine if it yields a maximum or minimum.

24. Slide 2.5 - 24 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Thus, x = 5 yields a maximum revenue. So, the theater should charge $26 – $5 = $21 per ticket.