Intro to Extrema: Videos & Practice Problems

Extrema on a graph include global extrema, which are the highest and lowest points of the entire function, and local extrema, which are the highest and lowest points within a specific region. A global maximum occurs when $f\left(c\right)>=f\left(x\right)$ for all $x$, while a local maximum is defined similarly but only for nearby points. Understanding these concepts is crucial for analyzing functions effectively.

Finding Extrema Graphically

Understanding extrema is crucial in analyzing the behavior of functions. Extrema refer to the maximum and minimum values of a function, and they can be categorized into two types: global (or absolute) extrema and local (or relative) extrema. Global extrema are the highest and lowest points of the entire function, while local extrema are the highest and lowest points within a specific region of the function.

To identify a global maximum, we consider a point \( c \) in the function. If \( f(c) \) is greater than or equal to \( f(x) \) for all \( x \) in the function, then \( c \) represents a global maximum. Conversely, a global minimum occurs at point \( c \) if \( f(c) \) is less than or equal to \( f(x) \) for all \( x \). For example, if the highest point of a function is at \( (-3, 5) \), we can state that the global maximum is 5 at \( x = -3 \). Similarly, if the lowest point is at \( (-5, -5) \), this represents the global minimum.

Local extrema, on the other hand, focus on a smaller section of the graph. A local maximum occurs at point \( c \) if \( f(c) \) is greater than or equal to \( f(x) \) for all nearby \( x \) values. For instance, if a point at \( (2, 24) \) is higher than all nearby points, it is a local maximum. A local minimum is identified when \( f(c) \) is less than or equal to \( f(x) \) for all nearby \( x \) values, such as a valley point at \( (1, 1) \) that is lower than its surrounding points.

It is important to note that some points can be both global and local extrema. For example, if a global maximum is also the highest point in its immediate vicinity, it qualifies as both. However, endpoints of a function can be global extrema but typically do not qualify as local extrema due to their nature as boundaries. This distinction can vary based on conventions used in different courses, so it is advisable to confirm the specific guidelines with your instructor.

In summary, recognizing the differences between global and local extrema enhances our understanding of function behavior, allowing for more effective analysis and interpretation of graphs.

Finding Extrema Graphically Example 1

In analyzing the extreme values of a function defined over various domains, it is essential to understand the concepts of global and local extrema. The function in question is examined across three different domains, leading to distinct conclusions about its extreme values.

For the first graph, where the function is defined over all real numbers (from negative infinity to positive infinity), the lowest point occurs at \( x = 0 \), yielding a value of \( f(0) = 0 \). This point is classified as both a global minimum and a local minimum since it is the lowest point in the entire domain and is not an endpoint. However, the function does not have a maximum value, as it extends infinitely in the positive direction.

In the second graph, the function is restricted to the domain from \( 0 \) to \( 3 \), including both endpoints. Here, the global minimum is again at \( x = 0 \) with \( f(0) = 0 \). However, this point is not considered a local minimum due to the convention that endpoints cannot be local extrema. The global maximum occurs at \( x = 3 \), where the function reaches a value of \( f(3) = 9 \). Thus, this function has a global minimum of \( 0 \) and a global maximum of \( 9 \), but no local extrema.

In the final graph, the domain is again from \( 0 \) to \( 3 \), but this time the endpoints are excluded. Consequently, while the function approaches \( 0 \) as closely as desired, it never actually reaches it, meaning there is no global minimum. Similarly, the function can get arbitrarily close to \( 9 \) but does not include it, resulting in no global maximum either. Therefore, this function has no extreme values within the specified domain.

These examples illustrate how the definition of the domain significantly impacts the identification of extreme values. Understanding the distinction between global and local extrema, especially in relation to endpoints, is crucial for accurately determining the behavior of functions across different intervals.

Finding Extrema Graphically Example 2

In analyzing the function f(x) for local and global maxima and minima, it is essential to identify the peaks and valleys on the graph. The highest point of the function occurs at x = -2, which is classified as both a global maximum and a local maximum, as it is the highest point in the entire function and also the highest among its immediate neighbors.

Conversely, the lowest point of the function is found at x = 1, where f(1) = -5. This point serves as both a global minimum and a local minimum, being the lowest in the entire function and among nearby points.

Further examination reveals additional local extrema. At x = -4, a local minimum is identified, as it is the lowest point in its vicinity. On the other hand, at x = 3, a local maximum is present, being the highest among nearby points. Additionally, at x = 5, another local minimum is observed, as it is lower than surrounding points.

It is important to note that endpoints of the function do not qualify as local extrema according to the conventions used in this analysis. Therefore, while we have identified all local and global maxima and minima, endpoints are excluded from being classified as extreme values.

Finding Extrema Graphically Example 3

In analyzing the graph of the sine function, we can identify the locations of local and global maxima and minima. The sine function oscillates between 1 and -1, creating multiple peaks and valleys. This characteristic can be confusing, especially when compared to functions with a single highest and lowest point.

To determine the global maxima, we observe that the highest value the sine function reaches is 1. This occurs at several points, notably at \( x = \frac{\pi}{2} \) and \( x = \frac{5\pi}{2} \). Both of these points are classified as global maxima because they represent the absolute highest value of the function. Additionally, they are local maxima since they are higher than the surrounding points and are not endpoints.

Similarly, the global minima of the sine function occur at the lowest value of -1, which is reached at \( x = \frac{3\pi}{2} \) and \( x = \frac{7\pi}{2} \). These points are also both global and local minima, as they represent the absolute lowest value of the function and are lower than their neighboring points.

It is important to understand that a function can have multiple global maxima and minima, particularly in oscillating functions like the sine function. Each occurrence of the maximum or minimum value still qualifies as a global extremum, reinforcing the idea that the function reaches these points multiple times throughout its domain.

Determine where the local and absolute maxima and minima occur on the given graph of $f\left(x\right)$.

• A. Absolute max of 4 at $x=-2$, Absolute min of $-3$ at $x=5$
• B. Absolute max of 4 at $x=-2$, Local max of 4 at $x=-2$, No absolute or local mins
• C. Absolute max of 4 at $x=-2$, Absolute min of 2 at $x=-5$
• D. Absolute max of 4 at $x=-2$, Local max of 4 at $x=-2$, Local min of 2 at $x=-5$

Answer: (B)

Determine where the local and absolute maxima and minima occur on the given graph of $f\left(x\right)$.

• A. Absolute max of 5 at $x=4$, Local max of 4 at $x=-3$, Absolute min of $-2$ at $x=1$, Local min of $-2$ at $x=1$ & 1.5 at $x=-4.5$
• B. Absolute max of 5 at $x=4$, Absolute min of $-2$ at $x=1$, No local extrema
• C. Absolute max of 4 at $x=-3$, Local max of 4 at $x=-3$, Absolute min of $-2$ at $x=1$, Local min of $-2$ at $x=1$
• D. Absolute max of 5 at $x=4$, Local max of 4 at $x=-3$, Absolute min of $-2$ at $x=1$, Local min of $-2$ at $x=1$

Answer: (D)

Finding Extrema Graphically Example 4

To sketch a graph of a function that is continuous over the interval from -3 to 2, including both endpoints, we need to incorporate specific properties regarding its maximum and minimum values. The function must have an absolute maximum at \( x = 2 \), an absolute minimum at \( x = 0 \), and a local maximum at \( x = -2 \).

First, we establish the endpoints of the graph at \( x = -3 \) and \( x = 2 \). Since these endpoints are included in the interval, they will be represented as solid dots on the graph. The absolute maximum at \( x = 2 \) indicates that this point will be the highest value of the function within the specified interval. Therefore, we can place this point at a higher value on the y-axis, ensuring it is the peak of the graph.

Next, we identify the absolute minimum at \( x = 0 \). This point will represent the lowest value of the function, so we position it lower on the y-axis, creating a valley or dip in the graph. Since this is not an endpoint, the graph will curve downwards to this point.

For the local maximum at \( x = -2 \), we need to create a peak that is higher than the values immediately surrounding it but lower than the absolute maximum at \( x = 2 \). This peak will be represented on the graph at \( x = -2 \), ensuring it does not exceed the height of the absolute maximum.

Finally, we include the endpoint at \( x = -3 \). This point will not be an extreme value, so we can place it on the graph without affecting the local or absolute extrema. The graph must be continuous, meaning there should be no breaks or holes as we connect these points.

In summary, the graph will start at the endpoint \( x = -3 \), rise to the local maximum at \( x = -2 \), drop down to the absolute minimum at \( x = 0 \), and then rise again to reach the absolute maximum at \( x = 2 \). This structure ensures that all specified properties are satisfied, resulting in a smooth, continuous function that adheres to the given conditions.

Here’s what students ask on this topic:

What is the difference between global extrema and local extrema?

Global extrema refer to the highest and lowest points of the entire function, known as the global maximum and global minimum. These points are determined by comparing the function's values over its entire domain. Local extrema, on the other hand, are the highest and lowest points within a specific region of the function. A local maximum occurs when the function's value at a point is greater than or equal to the values of nearby points, while a local minimum occurs when the function's value at a point is less than or equal to the values of nearby points. Understanding these concepts is crucial for analyzing functions effectively.

How do you identify a global maximum on a graph?

To identify a global maximum on a graph, you need to find the highest point of the entire function. This point occurs when the function's value at a specific x-value, f(c), is greater than or equal to the function's value at all other x-values in the domain. Mathematically, this is expressed as $f\left(c\right)>=f\left(x\right)$ for all x in the domain. On the graph, this point will be the highest point when considering the entire function.

Can a point be both a global and local extremum?

Yes, a point can be both a global and local extremum. If a point is the highest or lowest point of the entire function (global extremum) and also the highest or lowest point within a specific region (local extremum), it qualifies as both. For example, if a point is the global maximum, it is also a local maximum if it is higher than all nearby points. However, endpoints can be global extrema but not local extrema, depending on the convention used in your coursework.

What is the significance of endpoints in determining extrema?

Endpoints play a significant role in determining extrema, especially for global extrema. An endpoint can be a global maximum or minimum if it is the highest or lowest point of the entire function. However, by the convention used in some courses, endpoints cannot be local extrema because they do not have nearby points on one side. It is important to verify the convention used in your specific coursework or textbook to understand how endpoints are treated in determining extrema.

How do you identify a local minimum on a graph?

To identify a local minimum on a graph, you need to find a point where the function's value is lower than or equal to the values of nearby points. Mathematically, this is expressed as $f\left(c\right)<=f\left(x\right)$ for all nearby x-values. On the graph, this point will appear as a valley or the lowest point within a small region. By zooming in on the graph, you can identify these local minima by comparing the function's values at nearby points.