Overview
After our discussion of families of functions, we will now proceed to the topic of optimization. Our main focus will be on understanding how calculus ideas can be used to answer the question of “where is a given function greatest or least?” We will begin with a more theoretical discussion of optimization, learning about the Extreme Value Theorem, and from there we’ll proceed to more applied settings where we see how calculus provides rigorous answers to interesting questions.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

(Review) Find all critical numbers of a function \(f\).

Recall and understand the definition of global maximum and global minimum, as well as how these types of extreme values differ from relative maxima and minima.

Explain what the Extreme Value Theorem says and enumerate the threestage process outlined in that theorem for finding the global extreme values of a function on a closed interval.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Understand and apply the Extreme Value Theorem.

Given a continuous function on a closed interval, find the global extreme values of the function.

Apply the process of global optimization to an applied setting.
Resources

Do the Preview Activity for 3.3 (on WeBWorK if required by your teacher).

Watch the following videos, making yourself a list of at least 3 things you learned from the videos or your reading  and include this as part of your feedback:

Watch the following videos about global optimization on a closed interval (all videos created at CCSL):

Watch the following videos about global optimization on an open interval (all videos created at CCSL):

Watch the following video, which shows a detailed example of an optimization problem involving a family of functions (video created at CCSL):