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
- Application-Extrema
Exit problems of the session
-
Find the critical point in the domain of the following functions:
(a). $y=4\sqrt{x}-x^2$ (b). $y=\cos(2x)$ - Find the absolute maxima and absolute minima of the following function over the given interval.
(a). $y=x-x^2$ over $[-1. 1]$ (b). $y=x+\sin x$ over $[0, 2\pi]$
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 $c$ be an interior point in the domain of $f(x)$. We say that $c$ is a critical point of $f$ if $f'(c)=0$ or $f'(c)$ 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 $f(x)$ over a closed interval $[a, b]$:
- Evaluate $f$ at the end point $x=a$ and $x=b$.
- Find all critical points of $f$ that lie over $(a, b)$.
- Evaluate $f$ at those critical points.
- Compare all the values found in 1 and 3. The largest of these values is the absolute maximum of $f$, and the smallest of these values is the absolute minimum of $f$.
Videos and Practice Problems of Selected Topics
- $\rhd$ Introduction to minimum and maximum points (5:29) Find the absolute and relative extrema on a graph.
- $\rhd$ 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.
- $\rhd$ Finding critical points (5:50) Find the critical points of $f(x) = xe^{-2x^2}$.
- * Practice Find the critical points. (4 problems)
- $\rhd$ Extreme Value Theorem (7:57) A discussion on the Extreme Value Theorem.
- $\rhd$ Finding absolute extrema on a closed interval (6:55) Find the maximum value of $f(x) = 8\ln x-x^2$ over $[1,4]$.
- * 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?