Learning Outcomes

1. Define absolute extrema and local extrema
2. Find the critical points of a function over a closed interval.
3. 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

• Application-Extrema

Exit problems of the session

1. Find the critical point in the domain of the following functions:

(a). (b). 2. 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 :
1. Evaluate at the end point and .
2. Find all critical points of that lie over 3. Evaluate at those critical points.
4. 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

1. Introduction to minimum and maximum points (5:29) Find the absolute and relative extrema on a graph.
2. 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.
3. Finding critical points (5:50) Find the critical points of .
4. * Practice Find the critical points. (4 problems)
5. Extreme Value Theorem (7:57) A discussion on the Extreme Value Theorem.
6. Finding absolute extrema on a closed interval  (6:55) Find the maximum value of over .
7. * Practice Absolute minima and maxima over closed intervals. (4 problems)