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. \(f\) is continuous, but not necessarily differentiable, has domain \([0,6]\), and has one local minimum and no local maximum on \((0,6)\)

Short answer

Sketch a continuous curve with a local minimum and no maximum within \(x \in (0, 6)\).

Step-by-step solution

Understand the Requirements

We need to sketch a function that is continuous on the interval \([0, 6]\), meaning the graph cannot have breaks, gaps, or jumps. Furthermore, it has one local minimum and no local maxima within the open interval \((0, 6)\). This means exactly one point on the graph should be at the lowest curve within \((0, 6)\) and not at the endpoint.

Define the End Behavior and Critical Points

Since the function is continuous over \([0, 6]\) without differentiability conditions, we can use linear or piecewise components. There is no local maximum, so the graph should not increase after decreasing or vice versa around any critical points. The function should ideally start and end at different heights to allow a minimum inside \((0, 6)\).

Sketch the Function

Begin by plotting the interval from \(x=0\) to \(x=6\). Start at a height, say 3, at \(x=0\), then draw a downward curve reaching a local minimum at, for instance, \(x=3\). After the minimum, smoothly increase the height until the endpoint at \(x=6\). The height at \(x=6\) can be slightly different from the start, say 4, to maintain continuity and avoid a local maximum.

Verify the Conditions

Check that the graph is indeed continuous from \(x=0\) to \(x=6\) and contains no gaps. Ensure the function has exactly one turning point, a minimum, around, say \(x=3\), with no points where it increases then decreases within \((0, 6)\) to ensure no local maxima. Conclude that both criteria of the function are satisfied.

Key concepts

Continuous Functions

In calculus, a function is continuous if there are no interruptions in its graph. This means the graph can be drawn without lifting your pencil from the paper. At any point within the interval, a continuous function will not have any breaks, gaps, or jumps.

Characteristics of continuous functions include:

• Every real number within the domain of the function has an output, or in other words, is covered by the function.
• As you move along the graph, there are no abrupt stops or jumps.
• At any specific point, the function value and the limit of the function as you approach the point are the same.

For example, imagine a function that describes a smooth hill. As you walk up and down, you never step into thin air because the slope is consistently defined. These properties are crucial in understanding why continuous functions are so central in calculus. They ensure predictability and smoothness over a given domain.

Local Minimum

A local minimum is a specific point on the graph of a function where the function reaches its smallest value when compared to nearby points. In other words, it is a trough or a dip on the graph where the function goes down to a valley and then starts going up.

How to identify and understand local minima:

• They occur where the function changes direction from decreasing to increasing.
• A local minimum is not necessarily the lowest point of the entire graph, only the lowest in a small surrounding region.
• On the graph of a function, especially one limited to a certain interval, there could be one or more local minima depending on the graph’s behavior.

In the context of our exercise, the function is expected to have exactly one local minimum within the open interval (0, 6), indicating a clear point of the graph dipping before rising again. Identifying this minimum helps determine the general shape and flow of the graph.

Critical Points

Critical points of a function occur where the derivative is zero or undefined, leading to potential changes in the direction of the graph. These points are where the graph might peak, trough, or level off, making them vital for understanding the graph's shape.

Characteristics and importance of critical points:

• At critical points, the slope of the tangent to the graph is zero (indicating a potential peak or trough) or undefined (often indicating a cusp or vertical tangent).
• They help identify potential local maxima or minima but require further analysis to confirm such features.
• In planning to sketch a function, identifying critical points helps delineate where changes in direction or slope might occur, impacting the overall graph structure.
• For a continuous function that is not necessarily differentiable, like the one in the exercise, critical points guide where we design curves or angles to meet the given criteria of the function's behavior.

Recognizing and plotting these points help ensure the drawn graph fulfills all necessary conditions, such as having a single local minimum within a specified interval, without unexpected peaks or discontinuities.