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 differentiable, has domain \([0,6]\), and has two local maxima and two local minima on \((0,6)\).

Short answer

Yes, it's possible. The function requires smooth oscillations twice across \((0,6)\) while remaining differentiable.

Step-by-step solution

Understanding Differentiability

Since the function \( f \) is said to be differentiable, it means that the function is smooth with no sharp corners or jumps across its domain. This indicates the function's derivative \( f'(x) \) exists and is continuous on \( (0,6) \). We should keep this in mind when sketching, ensuring the graph appears smooth.

Analyzing the Domain

The function has the domain \([0,6]\). This means it is defined, including its endpoints at \( x = 0 \) and \( x = 6 \). However, the local extrema (maxima and minima) occur strictly between these two points, i.e., in the interval \((0,6)\). The function values at the endpoints are relevant but won't affect the nature of local extrema unless they are also end points of the interval.

Understanding Local Extrema

Two local maxima and two local minima indicate that the function \( f \) turns direction four times within the open interval \((0,6)\). A maximum implies the function rises to a peak and then falls, whereas a minimum implies it drops to a trough and then rises. This can be visualized as oscillating behavior resulting in a wavelike graph.

Sketching the Outline of the Graph

Create a rough sketch: start at some point \( x=a \) (where \( 0<a<3 \)), reach a local maximum, then descend to a local minimum at \( x=b \) (\( a<b<3 \)). Rise again to reach another maximum at \( x=c \) (\( 3<c<6 \)), and finally descend to a local minimum at \( x=d \) (\( c<d<6 \)).

Endpoints and Continuity Check

Decide on the behavior of the endpoints. They could either form higher/lower outer points compared to any pre-existing local extrema to avoid additional local extrema at the edges. Verify that the graph is continuous across \([0,6]\) and no additional local minima/maxima are introduced inadvertently.

Confirming Two Maxima and Minima

Ensure that there are precisely two peaks and two troughs by checking the sketched turning points corresponding to local maxima and minima, respecting the differentiability criterion.

Key concepts

Local Maxima

A local maximum is a point on a function where the function's value is higher than all other values in its immediate vicinity. It is essentially a peak within a specified interval. To detect a local maximum on a graph, observe where the function goes up to and then starts to descend. Here, the derivative of the function changes from positive to negative.

• The role of local maxima is crucial when sketching graphs, as they represent high points where the function changes direction.
• Analyzing a function for local maxima helps in understanding its oscillatory behavior, which can contribute to constructing an accurate graph.

In the given exercise, the presence of two local maxima in the interval (0, 6) means our function should reach peaks twice as it progresses horizontally. This contributes to the wavelike form of the graph.

Local Minima

A local minimum is a point on a function where the function's value is lower than all other values in its immediate surroundings. Visually, it represents a trough in the graph. Here, the derivative changes from negative to positive.

• Local minima, like local maxima, mark critical points where the function changes its path from descending to ascending.
• They are integral to identifying interval oscillations in functions, which inform us how dips occur between peaks.

In the step-by-step exercise, the requirement for two local minima means that our function should encounter downward rides into troughs twice within the (0, 6) interval. The presence of these points, along with local maxima, defines the essential wave shape.

Graph Sketching

Graph sketching involves visually representing a function to convey its behavior clearly. To sketch effectively, it is crucial to take into account differentiability, local extremas, and overall continuity.

• Ensure that the graph is smooth, reflecting the differentiable nature of the function – meaning no abrupt changes or sharp points.
• Identify the peaks and troughs accurately to mark local maxima and minima positions on the graph.
• Consider the domain, particularly endpoints, to align the graph's spread appropriately between defined limits.

Given the exercise, sketching a function that oscillates between two local maxima and minima within the domain requires calculating where these points possibly lie across the function's continuous path. This kind of sketch depicts the overarching wave-like pattern required by the problem's constraints.

Continuity of Functions

Continuity of a function is a fundamental concept related to how the function behaves within its domain. A continuous function has no breaks, jumps, or gaps in its graph – it can be drawn without lifting the pencil.

• A continuous function is pivotal as it ensures predictability and smooth connections between local maxima and minima.
• Checking endpoints and behavior within the domain helps ensure the function's smooth transition from start to finish.
• The differentiable nature implies continuity more strongly on the interior of the domain, guaranteeing no interruptions in course or sudden jumps.

In the context of sketching a graph with given local maxima and minima, ensuring continuity means confirming no unintended local extrema occur, and the ends of the domain connect seamlessly with the depicted graph. This verifies the representation remains true to analyzed behavior throughout (0, 6).