Definition 3.1.1. Absolute Maximum.
A value \(c \in [a,b]\) is an absolute maximum of a function \(f\) over the interval \([a,b]\) if and only if \(f(c) \ge f(x)\) for all \(x \in [a,b]\text{.}\)
Section 3.1 Absolute and Relative Extrema
Standards
- Interpret derivatives in an application
- Use a derivative in an application
- Analyze a function using derivatives
- Find extrema using derivatives
Note in this section we do not finish these standards but rather provide the initial understanding. These standards are completed in later sections.
One of the uses of derivatives is to identify traits of functions such as highest and lowest points. These are optimizations in many applications (i.e., find the lowest cost or maximum profit).
Our goals are to understand what these traits are, identify where they can occur, then determine how to reliably find these locations.
Subsection 3.1.1 Absolute Extrema
First, we will think about traits of highest and lowest points on specified sections of curves (functions over a limited domain).
An absolute maximum is the highest point of a function/curve on a specified interval. An absolute minimum is the lowest point of a function/curve on a specified interval. Collectively maxima and minima are known as extrema.
Definition 3.1.2. Absolute Minimum.
A value \(c \in [a,b]\) is an absolute minimum of a function \(f\) over the interval \([a,b]\) if and only if \(f(c) \le f(x)\) for all \(x \in [a,b]\text{.}\)
Activity 3.1.1. Identifying Absolute Extrema.
The goal of this activity is to determine where we should look for absolute extrema.
(a)
Identify where (x values) the absolute maxima and absolute minima occur for each function in Figure 3.1.3 in the intervals specified. Note these are closed intervals (the endpoints are included). The intervals are specified by the green, dashed, vertical lines.
(b)
Using these examples determine where on a function we should look for absolute extrema. For now we are not interested in calculus ideas. Simply describe what is happening where the absolute extrema occur. Avoid using ‘highest/lowest’. This description is re-iterating the definition and relies on visual interpretation which is imprecise (can you tell if it is \(x=0.235\) or \(x=0.224\text{?}\)).
(c)
Construct another absolute extrema practice problem for another group to solve.
Subsection 3.1.2 Identifying Relative Extrema
A relative maximum is a location on a curve where all points near it are lower. A relative minimum is a location on a curve where all points near it are higher.
Definition 3.1.4. Relative Maximum.
A value \(c\) in the domain of a function \(f\) is a relative maximum of \(f\) if and only if there exists some interval \((a,b)\) in the domain containing \(c\) such that \(f(c) \ge f(x)\) for all \(x \in (a,b)\text{.}\)
Definition 3.1.5. Relative Minimum.
A value \(c\) in the domain of a function \(f\) is a relative minimum of \(f\) if and only if there exists some interval \((a,b)\) in the domain containing \(c\) such that \(f(c) \le f(x)\) for all \(x \in (a,b)\text{.}\)
Activity 3.1.2. Understanding Relative Extrema.
The goal of this activity is to understand the definition, specifically the part of an interval existing. Reference Figure 3.1.6.
(a)
There is a relative maximum near \(x=5\text{.}\) Write an interval in which that point is higher than all the other points in your selected interval.
(b)
There is another relative maximum near \(x=0.4\text{.}\) Write an interval in which that point is higher than all the other points in your selected interval.
(c)
How does the selection of interval enable there to be more than one maximum?
Activity 3.1.3. Identifying Relative Extrema.
The goal of this activity is to determine where we should look for relative extrema.
(a)
Identify the relative maxima and relative minima for each function in Figure 3.1.7.
(b)
Draw the tangent line to the curve at each relative extreme point. What is the slope of these tangents? How is slope of a tangent calculated?
(c)
Using these examples develop a method for finding relative extrema.
Activity 3.1.4. Conditions for Extrema.
The goal of this activity is to discover properties of curves that have specific types of extrema.
(a)
Draw one curve for each of the following sets of conditions.
- One relative maximum, two relative minima
- One relative minimum, two relative maxima
- One relative minimum, no relative maxima
- Two relative minima, no relative maxima
- One absolute maximum, one absolute minimum
- One absolute maximum, two absolute minima
- One absolute maximum, two relative maxima
- No absolute maximum
(b)
Conjecture conditions for when absolute extrema must exist.
