- 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.
- Chapter 4.3 Maxima and Minima
- p. 376: 108, 110, 113, 119, 122, 124
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
- 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)
- How to use a spring to calculate mass in outer space
- True/False: How reliable are your antibody tests?