# Optimization The purpose of **optimization** is to find the best value with regards to a given set of criterion. This is likely one of the **most widely used applications of calculus** within the real-world and is an invaluable skill to learn, There are generally two parts to these problems: 1. The **Constraint Function(s)**: aaa 2. The **Objective Function(s)**: aaa ## Applications of Optimization There are countless situations involving quantities that must be optimized: | Unit | Example | | --- | --- | | Volume | *Manufacturing a tin can that holds the largest amount of a liquid* | | Distance | *Creating a delivery route to minimize the mileage on a vehicle* | | Area | *Designing a garden with the most room for growing vegetables* | | Surface Area | *Creating a box with the least amount of material* | | Perimeter | *Enclosing a horse pen with the lowest cost of fencing material* | ## Optimization Practice The following is a general strategy that can be followed when solving optimization problems: > [!hint]+ Strategy: Solving Optimization Problems > 1. *Introduce all variables.* If applicable, draw a figure and label all variables. > > 2. Determine *which quantity is to be maximized or minimized*, and for what *range of values* of the other variables (if this can be determined at this time). > > 3. Write a formula for the quantity that needs to be maximized/minimized in terms of the other variable(s). > > 4. Write any equation relating the independent variables in the formula from step 3. Use these equations to write the quantity to be maximized/minimized as a function of one variable. > > 5. Identify the domain of consideration for the function in step 4 based on the physical problem to be solved. > > 6. Locate the maximum or minimum value of the function from step 4. This step typically involves looking for critical points and evaluating a function at endpoints.