1. Clicker Question 1 (0, 1) is a critical point for the function f (x ) = x 4 – 4x 3 + 1. This point is a A. local maximum B. local minimum C. neither

2. Clicker Question 2 On what interval is that same function f (x ) = x 4 – 4x 3 + 1 concave down? A. (- , 0] B. (- , 2] C. [0, 2] D. [0, 3] E. [2,  )

3. Optimization and Modeling (4/3/09) Optimizing functions is pretty easy when you are given an explicit formula to work with (just check critical points, endpoints, and end behavior). More demanding is when we must mathematically model a situation, i.e., we must develop the function first, then analyze it.

4. An example with no modeling required Find the exact global (absolute) maximum and minimum values of f (x ) = x 3 – 3x + 2 on the interval [-2.5,1.5].

5. An example with just a little modeling Find two numbers whose sum is 30 and whose product is maximal. Be sure to check that you have really found a maximum.

6. An example of modeling Suppose we want to produce an open- topped box with a square base which is to hold 4000 cubic inches. What dimensions of the box will use the least amount of cardboard material? How much cardboard is needed? (Assume no overlap.) Make sure your answer is really a minimum.

7. Tips on Optimization Modeling Ask: What quantity am I optimizing? Draw and label a sketch (if possible). Write a formula for the optimized quantity in terms of the necessary variables. Use any given constraints to eliminate all but one independent variable. Compute critical points and evaluate them and endpoints to get global maxima and/or minima. Make sure you have answered the original question!

8. Assignment For Monday, read Section 4.7 (we will skip 4.4 - 4.6) and do Exercises 1, 3, 5, 7, 9, 13, 17, 19. Hand-in #3 is due Tues 4/7 at 4:45 pm. Test #2 is on Monday, April 13. We will have a short (optional) class on Fri 4/10 but a longer class on Monday 4/13.