Question

Sketch a graph of a function with the given properties. If it is impossible to graph such a function, then indicate this and justify your answer. \(f\) has domain \([0,6]\), but is not necessarily continuous, and has two local maxima and no local minimum on \((0,6)\).

Short answer

Graph a function with peaks at two places between 0 and 6, without dips that act as local minima.

Step-by-step solution

Understand the Domain

The function is defined on the domain \([0, 6]\). This indicates that the function exists for all values of \(x\) between and including 0 to 6. This means we need to ensure any graph or function does not extend beyond these points.

Consider Local Maxima

We need to have two local maxima within the open interval \((0, 6)\). A local maximum is a point where the function changes from increasing to decreasing. We need to envision two such peaks in the graph.

Address Continuity

The function is not necessarily continuous, meaning it can be discontinuous at some points within the domain. However, for simplicity, we will initially assume it is a piecewise continuous function since the problem does not disallow it.

Sketch a Sample Function

Draw the x-axis from 0 to 6. Pick two points within this interval, say \(x = 2\) and \(x = 4\), and plot them as local maxima. The graph might increase to these points and then decrease immediately afterward, with no other ups and downs until \(x = 6\).

Ensure No Local Minima

To ensure there are no local minima, ensure that between our local maxima peaks, the function does not dip and rise sharply creating another minimum. Instead, it can flatten or connect directly to endpoints.

Key concepts

Domain of a Function

The domain of a function is a crucial aspect to understand when sketching or analyzing a graph. In the simplest terms, it refers to all the possible values of the independent variable, often represented as "x," for which the function is defined. In this exercise, the function has a domain of \([0, 6]\), meaning our function exists for every \(x\) value between and including 0 and 6.

This domain definition allows us to confidently sketch or construct a graph within this boundary. Outside these points, the function does not exist, and hence, the graph should not extend beyond them. Remember:

• The endpoints, in this case, 0 and 6, are part of the domain.
• Take care of how the function behaves at these endpoints.
• Always verify if there are any additional constraints that could affect the domain.

Local Maxima

Local maxima are specific points on a graph where the function reaches a peak relative to its immediate neighbors. It represents a transition from increasing to decreasing behavior in the function's value.

To identify a local maximum, imagine a small "hill" on the graph. In this exercise, we need to ensure there are exactly two local maxima within the open interval \((0, 6)\).

Key aspects of local maxima include:

• The graph rises to reach these points.
• It falls afterward, forming peak-like structures.
• Local maxima can exist in continuous as well as piecewise continuous functions.

Piecewise Continuous Function

Piecewise continuous functions are those that are composed of multiple sections, each of which is continuous within its interval, though the function might not be continuous across the entire domain.

In this problem, even though the function is not necessarily continuous across \([0, 6]\), assuming it as piecewise continuous can simplify our approach. This allows for certain sections or intervals to have discontinuities, while maintaining continuity within smaller pieces.

Characteristics of piecewise continuous functions:

• The function is broken into "pieces" with their own continuous spans.
• Discontinuities can occur only where different pieces meet.
• They provide flexibility in modeling complex behaviors within a limited domain.

Graphical Representation

Creating a graphical representation of a function involves plotting its behavior across its domain. The function's criteria guide the graph's shape, slope, and features such as peaks (local maxima) or plateaus.

Let's take our specific case to illustrate:

• The x-axis ranges from 0 to 6.
• Two distinct peaks or local maxima are necessary in the interval \((0, 6)\), ensuring the function's compliance with the exercise requirements.
• No local minima should appear between these maxima.
• The function may show flat regions or straight line segments as connections, especially between the local maxima, to avoid forming additional peaks or troughs.

Discontinuity in Functions

Discontinuity in functions refers to points within the domain where the function is not continuous. This could mean jumps, holes, or any kind of break in the graph. In this problem, the function is described as potentially discontinuous, meaning there might be points where the function abruptly changes its value.

Here's how discontinuity might be handled:

• If the function is piecewise continuous, we might see abrupt changes at "junctions."
• These discontinuities don't prevent the determination of local maxima.
• It's crucial to manage discontinuities within the domain without disrupting the required properties, like the number of local maxima in this case.

Understanding discontinuities is essential as they significantly affect the shape and properties of the graph.