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 three local maxima and no local minimum on \((0,6)\).

Short answer

The graph is possible: sketch three peaks at \( x = 1, 3, 5 \) with no valleys.

Step-by-step solution

Understand the Domain

The function \( f \) has a domain from \( 0 \) to \( 6 \). This means the graph of the function will only exist between these \( x \)-values, inclusive. The function might not be continuous, indicating possible discontinuities like jumps or breaks.

Identify Local Maxima Requirements

The function \( f \) must have three local maxima within the open interval \((0,6)\). A local maximum is a point where the function value is greater than values at nearby points. Importantly, there are no local minima specified, which means the function should only peak and not form any local dips.

Sketching the Function

To sketch this function with three local maxima and no local minima, begin by plotting \( x \)-values within \( 0 \) and \( 6 \). Imagine peaks at three distinct points, such as \( x = 1, x = 3, \) and \( x = 5 \). Draw curves rising to peaks at these points. Avoid creating any valley or dip between these peaks.

Consider Discontinuities

Since continuity is not required, you could introduce discontinuities. For example, right after a peak, the function might jump down vertically and rise again to another peak, mimicking certain behaviors seen in piecewise functions.

Validate the Sketch

Ensure the final sketch includes exactly three local maxima without any other turning points or local minima within the domain. Check both ends of the domain \( [0,6] \) for adherence to all given conditions.

Key concepts

Understanding Local Maxima

When exploring functions, a local maximum holds significant importance. A local maximum is a point where the function's value is the highest compared to its neighbors. Imagine climbing a small hill. The top of the hill, where you stop climbing and start descending, represents a local maximum. These maxima can occur at different points within a domain, offering peaks in the function's graph.

In terms of graphing, a local maximum does not have to be the absolute highest point on the function. It just needs to be higher than the points surrounding it. For example, in our exercise, the function needs three of these local maxima within the interval \(0,6\). Each peak

• is separate and distinct, ensuring the graph shows clear individual peaks,
• does not lead to a valley or dip, maintaining a consistent climb towards each maximum,
• represents a point that quickly descends or flattens out after reaching the peak.

These conditions are crucial to creating a function that visually matches the specified requirements.

Defining Domain and Range

The domain and range of a function determine where the graph can exist along the coordinate plane. Think of the domain as all the possible x-values (inputs) that a function can have. The range, on the other hand, represents all the possible y-values (outputs) the function can produce.

In our task, the function is defined over the domain \[0,6\]. This means the graph of the function exists strictly between \(x = 0\) and \(x = 6\), inclusive. All elements or phenomena within the function will occur within this window. When sketching or looking at this function:

• Expect no values of \(x\) outside this range. The graph won't extend beyond these bounds as per the problem's constraints.
• Check your function’s behavior at \(x = 0\) and \(x = 6\) to ensure it respects the domain's limits.

The absence of a range specification means we can focus on the behavior, like peaks and discontinuities, rather than worrying about specific y-values.

Spotting and Understanding Discontinuities

Discontinuities occur when a graph of a function has breaks or jumps. This behavior is not continuous, meaning the graph doesn't smoothly connect throughout the defined domain. Think of a rally car jumping over small gaps in the road while racing. These jumps are the discontinuities in our graphing exercise.

In practical terms, discontinuities happen when:

• The function abruptly jumps from one value to another, without drawing a continuous line between them.
• There are holes or gaps at certain points in the graph, where the function is not defined.
• The graph suddenly spikes up or drops down, often seen in piecewise functions.

To showcase discontinuities skillfully in our function, consider introducing jumps after reaching each local maximum, indicating a reset point before rising again to next peak. This way, the graph doesn't form valleys that would suggest a local minimum, adhering to our exercise's constraints.