Learning Outcomes
 Define absolute extrema and local extrema
 Find the critical points of a function over a closed interval.
 Use critical points to locate absolute extrema over a closed interval.
Textbook
 Chapter 4.3 Maxima and Minima
Textbook Assignment
 p. 376: 108, 110, 113, 119, 122, 124
WeBWork Assignment
 ApplicationExtrema
Exit problems of the session

Find the critical point in the domain of the following functions:
(a). (b).  Find the absolute maxima and absolute minima of the following function over the given interval.
(a). over (b). over
Key Concepts
 A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum.
 Let be an interior point in the domain of . We say that is a critical point of if or is undefined.
 If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.
 A continuous function over a closed interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint.
 Steps of locating absolute extrema of over a closed interval :
 Evaluate at the end point and .
 Find all critical points of that lie over .
 Evaluate at those critical points.
 Compare all the values found in 1 and 3. The largest of these values is the absolute maximum of , and the smallest of these values is the absolute minimum of .
Videos and Practice Problems of Selected Topics
 Introduction to minimum and maximum points (5:29) Find the absolute and relative extrema on a graph.
 Critical points (7:52) The “critical points” of a function are defined followed by a discussion on their relationship with the extrema of the function.
 Finding critical points (5:50) Find the critical points of .
 * Practice Find the critical points. (4 problems)
 Extreme Value Theorem (7:57) A discussion on the Extreme Value Theorem.
 Finding absolute extrema on a closed interval (6:55) Find the maximum value of over .
 * Practice Absolute minima and maxima over closed intervals. (4 problems)
STEM Applications
 How to use a spring to calculate mass in outer space
 True/False: How reliable are your antibody tests?