Optimization Worksheet When we say we want to optimize something, we typically mean, find the absolute maximum or minimum. 1. Given a function x) on a restricted interval [(1, b], what steps do you need to take in order to find the absolute maximum and absolute minimum value? 2. Find the absolute maximum and minimum values off(x) = x3 + 6.5x2 + 4x on the interval [2, 1]. 3. What would you do if you wanted to verify that the absolute minimum you found is indeed a minimum? 4. Use the first or second derivative test to verify that the minimum you found above is indeed a minimum. Suppose that you have a piece of cardboard that is 2m by 4m. You want to fold this into a box without a top by cutting little squares out of the corners and folding up the edges. eaa Goal: Find what height you should make the box in order to maximize the volume ofthe box. (Note: another way to say this is, how big should the little corner pieces you cut out be?) 5. Use the sketch ofthe cardboard above to help you sketch the box. a. The height of the box is m b. The width of the box is m c. The length of the box is m d. The volume of the box is m 6. Find the critical numbers for yourvolume function. 7. Find the endpoints in this scenario. a. What is the smallest amount you could cut out from the corners? b. What is the largest amount you could cut out from the corners? 8. Among your critical numbers and your endpoints, what value ofx will maximize your volume? 9. Use the first or second derivative test to verify that this is indeed a maximum, rather than a minimum. 10. What is the maximum volume of the box