1. 6.2: Applications of Extreme Values Objective: To use the derivative and extreme values to solve optimization problems.

2. In optimization problems, we are looking for the largest or smallest value a function can take. General Procedure for Optimization Problems: 1.Draw a diagram, if possible, and label the various unknown quantities that you will need. Figure out what needs to be maximized or minimized. 2.Write down the information given that is true regardless of the situation. Usually it is a fixed value or some sort of constraint. Write an equation to model this info. 3.Write a formula for what you are looking to maximize or minimize. 4.Rewrite your equation from step 2 in terms of a single variable. Substitute this expression into equation from step 3. 5.Be sure to check to see if there are any restrictions on the domain. These restrictions could indicate a closed interval. 6.Find the extreme values of the function that needs to be maximized or minimized.

3. Useful Formulas to Review Rectangular Prism: SA= 2lw + 2lh+ 2wh V=lwh Distance Formula: Cylinder: SA = 2πr 2 + 2πrh V= πr 2 h

4. EXAMPLES: 1. We need to enclose a rectangular field with a fence. We have 500 ft of fencing material and a building is on one side of the field so we won’t need any fencing. Determine the dimensions of the field that will enclose the largest area.

5. 2. We want to construct a box whose base length is 3 times the base width. The material used to build the top and bottom cost $10/ft 2 and the material used to build the sides cost $6/ft 2. If the box must have a volume of 50ft 3, determine the dimensions that will minimize the cost to build the box.

6. 3. A window is being built and the bottom is a rectangle and the top is a semicircle. If there is 12 meters of framing material, what must the dimensions be to let in the most light?