Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of a function with the given properties. \(f\) is continuous, but not necessarily differentiable, has domain \([0,6]\), reaches a maximum of 6 (attained when \(x=5\) ), and a minimum of 2 (attained when \(x=3\) ). Additionally, \(x=1\) and \(x=5\) are the only stationary points.

Step-by-Step Solution

Answer

Sketch a continuous curve from \(x=0\) to \(x=6\) with peaks at \(x=5\), troughs at \(x=3\), and plateaus at \(x=1\) and \(x=5\).

Step 1: Establish the Domain

The function \(f\) is defined over the domain \([0,6]\). This implies the graph of \(f\) will only exist for \(x\) values within this interval, including the endpoints \(x=0\) and \(x=6\).

Step 2: Identify Stationary Points

Stationary points occur where the derivative is zero (though the function isn't necessarily differentiable). We are given that \(x=1\) and \(x=5\) are the only stationary points. The graph will have horizontal tangents or plateauing behavior at these \(x\) values.

Step 3: Locate and Mark Extrema

The function has a maximum value of 6 at \(x=5\) and a minimum value of 2 at \(x=3\). On the graph, it should be clear that the function reaches its highest point at \(x=5\) and its lowest point at \(x=3\).

Step 4: Sketch the Graph from Endpoint to Endpoint

Starting from \(x=0\), draw the function continuously through \(x=1\) (stationary point), passing through the minimum at \(x=3\), and reaching the maximum at \(x=5\) (also a stationary point), then continue to \(x=6\). Ensure the graph goes from lower near the starting point \(y=0\) to higher towards \(x=5\), and back down slightly at \(x=6\).

Step 5: Verify Continous Behavior

Ensure the graph is continuous with no breaks, jumps, or gaps from \(x=0\) to \(x=6\). Despite the absence of differentiability on the entire domain, the graph should flow without interruption through all specified key points.

Key Concepts

Stationary Points

Stationary points are crucial features in graphing functions. They are the values of \(x\) where the derivative of a function equals zero, implying no change in the function's slope at these points. This gives rise to either horizontal tangents or a plateau on the function's graph, which signifies potential peaks, valleys, or points of inflection.
When you think about a function having stationary points, imagine them as resting places in the terrain of a graph. For the exercise at hand, it's important to note that \(x = 1\) and \(x = 5\) are the only stationary points given. The behavior at these points implies the graph will have a flat or leveled appearance at these x-values. However, keep in mind while discussing stationary points that the function need not be differentiable elsewhere.
Use these points to understand where the graph might change its slope or direction. They don't automatically indicate a maxima or minima, but they can strongly suggest such points when other conditions like extrema are also considered.

Extrema in Calculus

The concepts of extrema—maximum and minimum values—in calculus are pivotal in understanding a graph's general shape or behavior over its domain. Extrema serve as definitive highs and lows in the function's progression over a given range, which are crucial to determining the function's overall result or impact.
In this problem, the global maximum is specified at \(x = 5\) with a value of 6, and the global minimum is at \(x = 3\) with a value of 2. These are the highest and lowest function values over the interval \([0,6]\), and marking these points clarifies significant plot features.

• The maximum at \(x=5\) indicates where the function achieves its greatest y-value, helping visualize the peak of the graph.
• The minimum at \(x=3\) implies the lowest point within the defined interval, representing a trough on the graph.

By organizing the graph using these extrema, you ensure a clear understanding of where the graph rises and falls, resulting in a cohesive structure to the entire plot.

Continuous Functions

A continuous function forms the basis for smooth graph transitions with no breaks, jumps, or abrupt changes. Understanding a function as continuous means confirming it holds a reliable, steady path across its domain without sudden interruptions or separations.
The exercise specifies that the function \(f\) is continuous over the interval \([0,6]\). This means every value of \(x\) from 0 to 6 maps to a corresponding \(y\) point, ensuring the graph never fragments into unattached segments. Consider these key points when analyzing continuous functions:

• There should be a seamless connection between all points along the graph, ensuring no gaps.
• Continuous functions allow calculus operations—like integration over a range or finding limits—to function correctly.
• Even when functions are not differentiable at every point, continuity preserves the unity of the function's representation.

A solid understanding of continuity reassures that graphed functions appear true to their theoretical structure, thus enabling further mathematical explorations or evaluations if necessary.