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). $y=4\sqrt{x}x^2$ (b). $y=\cos(2x)$  Find the absolute maxima and absolute minima of the following function over the given interval.
(a). $y=xx^2$ 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 $f(x) = 8\ln xx^2$ 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?