1. Copyright © Cengage Learning. All rights reserved. 12 Further Applications of the Derivative

2. Copyright © Cengage Learning. All rights reserved. 12.2 Applications of Maxima and Minima

3. 33 Example 1 – Minimizing Average Cost Gymnast Clothing manufactures expensive hockey jerseys for sale to college bookstores in runs of up to 500. Its cost (in dollars) for a run of x hockey jerseys is C(x) = 2,000 + 10x + 0.2x 2. How many jerseys should Gymnast produce per run in order to minimize average cost?

4. 44 Example 1 – Solution Here is the procedure we will follow to solve problems like this. 1. Identify the unknown(s). There is one unknown: x, the number of hockey jerseys Gymnast should produce per run. 2. Identify the objective function. The objective function is the quantity that must be made as small (in this case) as possible.

5. 55 Example 1 – Solution In this example it is the average cost, which is given by 3. Identify the constraints (if any). At most 500 jerseys can be manufactured in a run. Also, is not defined. Thus, x is constrained by 0 < x ≤ 500. Put another way, the domain of the objective function is (0, 500]. cont’d

6. 66 Example 1 – Solution 4. State and solve the resulting optimization problem. Our optimization problem is: subject to 0 < x ≤ 500. We first calculate We solve to find x = ±100. We reject x = –100 because –100 is not in the domain of (and makes no sense), so we have one stationary point, at x = 100. There, the average cost is per jersey. cont’d Objective function Constraint

7. 77 Example 1 – Solution The only point at which the formula for is not defined is x = 0, but that is not in the domain of so we have no singular points. We have one endpoint in the domain, at x = 500. There, the average cost is cont’d