1. In this section, we will review the optimization process and begin using it in “real world” problems.

2. If f is a function that is continuous on [a, b], then f will achieve both an absolute maximum value and an absolute minimum value somewhere in [a, b].

3. Where, specifically, will we find these?

4. A function continuous on [a, b] can only achieve its absolute maximum and absolute minimum value at a critical point or at one of the endpoints x = a or x = b.

5. Find the absolute maximum and absolute minimum of on the interval [-1, 10].

6. This approach is basically the same when we are doing a “real world” optimization problem. The only differences are we typically will have to “create” the function that is to be optimized based on the information given. Also, instead of an interval [a, b], often we will have a constraint equation (a given relationship between the variables that must be “obeyed”).

7. Find the point on the curve that is closest to the point (1, 4).

8. A cylindrical can is to have total volume 1000cm 3. Find the dimensions of the can that would minimize the amount of material used to make the can.

9. A man launched his boat from point A on the bank of a river that is 3km wide. He wants to reach point B which is 8km downstream and on the opposite bank as quickly as possible. If he can row at 6km per hour and he can run 8km per hour, where should he land his boat to accomplish his goal?

10. A steel pipe is being carried down a hallway that is 9 feet wide. At the end of the hallway is a right angle turn into a hallway that is only 6 feet wide. What is the length of the largest pipe that can be carried horizontally around this corner?