4.4 Optimization Buffalo Bills Ranch, North Platte, Nebraska Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts Photo by Vickie Kelly, 1999

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose? There must be a local maximum here, since the area at the endpoints = 0. domain: x > 0 and x > 0 x < 20

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

To find the maximum (or minimum) value of a function: 1Write the OPTIMIZED function, restate it in terms of one variable, determine a sensible domain. 2Find the first derivative, set it equal to zero/ undefined, find function value at those critical points. 3 Find the function value at each end point.

Example 5: What dimensions for a one liter cylindrical can will use the least amount of material? We can minimize the material by minimizing the cans surface area. area of lids lateral area To rewrite using one variable, we need another equation that relates r and h :

Example 5: What dimensions for a one liter cylindrical can will use the least amount of material? area of ends lateral area

If a domain endpoint could be the maximum or minimum, you have to evaluate the function at each endpoint, too. Reminders: If the function that you want to optimize has more than one variable, find a second connecting equation and substitute to rewrite the function in terms of one variable. To confirm that the critical value youve found is a maximum or minimum, you should evaluate the function for that input value.