In this module you will find the extreme values of a function and solve three application problems that ask you to find either a maximum or a minimum. These problems were chosen because they illustrate the relationship between a function and its derivatives.

Problems that ask for maximum or minimum values are called optimization problems, and each solution usually requires a multi-step procedure. This procedure can often be summarized as follows:

1. Read the problem carefully and note which quantities vary and which are constant.
2. Draw a diagram. Label the components with appropriate variables and constants.
3. Identify the quantity to be maximized or minimized and write a function for it.
4. Find the domain of the function in the problem.
5. Identify critical points and any endpoints, which are the candidates for any maximum and minimum values of a function.
6. Use the first or second derivative to determine local maximum and minimum values.
7. Answer the original question. Does your answer make sense? Is it justified?

It is easy to get into in the individual steps and lose sight of this lengthy overall process. Computer algebra can help you focus on the overall process by assisting with the individual steps. This is particularly true when some of the individual steps are impractical without technology.

Lesson Index:

11.1 - Finding Extreme Values of a Function

11.2 - The Cone Problem

11.3 - The Can Problem

11.4 - The Can Problem With Waste

After completing this module, you should be able to do the following:

• Find the extreme values of a function
• Solve optimization problems using the procedure described above
• Use the TI-89 to define a function for the quantity to be optimized
• Use the graph of the first derivative and the value of the second derivative to justify your choice of maximum or minimum