Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of a function with the given properties. \(f\) is differentiable, has domain \([0,6]\), reaches a maximum of 4 (attained at two different values of \(x\), neither of which is an end point), and a minimum of 1 (attained at three different values of \(x\), exactly one of which is an end point.)

Step-by-Step Solution

Answer

The graph peaks at \(x=2\) and \(x=4\) with a value of 4, while valleys at \(x=0, 3,\) and \(6\) with a value of 1.

Step 1: Analyze the given properties

We know that the function \( f(x) \) is differentiable, meaning it is continuous and smooth. It has a domain of \([0, 6]\), meaning it is only defined within these bounds. The maximum value is 4, attained at two different \( x \) values that are not endpoints, and the minimum value is 1, attained at three different \( x \) values, precisely one of which is an endpoint.

Step 2: Determine x-values for Maximum and Minimum

From the properties, choose the following plausible x-values:- Maximum: the function reaches a value of 4 at \( x = 2 \) and \( x = 4 \).- Minimum: the function reaches a value of 1 at \( x = 0 \) (endpoint), \( x = 3 \), and \( x = 6 \) (endpoint).

Step 3: Sketch the Graph with Critical Points

Start plotting these points on the graph: - At \( x = 2 \) and \( x = 4 \), \( f(x) = 4 \).- At \( x = 0, 3, \) and \( 6 \), \( f(x) = 1 \).Ensure that the graph is smooth and differentiable, passing through these points.

Step 4: Connect the Points to Define the Shape

Create a smooth curve that: - Starts at the endpoint \((0,1)\).- Rises smoothly to the first maximum \((2,4)\).- Descends to the local minimum \((3,1)\).- Rises again to the second maximum \((4,4)\).- Finally descends to the endpoint \((6,1)\). Ensure the curve is differentiable at all points.

Key Concepts

Function Graphing

Graphing a function helps visualize how values of the function change over its domain. When graphing, we start by identifying important points known as critical points, which affect the graph's shape. Here, we're given specific properties like maximum and minimum values and a fixed domain from 0 to 6. The exercise involves creating a smooth, continuous line that reflects these constraints.

• The function is defined only within the interval [0, 6].
• Since the function is differentiable, there are no sharp corners or breaks.
• The endpoints and critical points provide guideposts for the graph's shape.

Begin plotting the known points: (0,1), (2,4), (3,1), (4,4), and (6,1). Connecting these with a smooth curve gives a visual representation of the function.

Critical Points

Critical points are crucial in understanding the behavior of a function. These are points where the function's derivative is zero or undefined, indicating possible maxima, minima, or changes in direction. For our function:

• The maximum values occur at critical points (2,4) and (4,4).
• The minimum values occur at (0,1), (3,1), and (6,1).

Unlike the maximum values, one of the minima occurs at an endpoint. This suggests that the function may start or end at an edge of the domain with a low value. By understanding critical points, you can predict where a function might rise to a peak or drop to a valley, crucial for sketching out the function's graph.

Maximum and Minimum Values

Maxima and minima represent the peaks and valleys of a function. They are essential in determining the overall shape and behavior of the graph. In this problem, the function:

• Reaches a maximum of 4 at two non-endpoint positions, (2, 4) and (4, 4).
• Achieves a minimum of 1 at three positions: (0, 1), (3, 1), and (6, 1), with (0, 1) and (6, 1) being endpoints.

These values guide how we draw the graph. You can visualize the curve climbing to 4, dropping to 1, climbing again, then finally dropping. These features help identify turning points in the graph, and understanding this enables you to accurately preview how the function behaves across its domain.

Endpoint Behavior

Endpoint behavior examines how a function acts at the boundaries of its domain. For a function defined on [0, 6], we carefully evaluate what happens at both x = 0 and x = 6:

• The function starts at (0, 1), indicating a minimum at this boundary.
• It ends at (6, 1), again at a minimum.

When endpoints are critical points or minimum/maximum points, they significantly frame the entire graph. In such cases, checking differentiability is necessary.
The smooth connection of these points without any jumps or cusps reflects the function's differentiable nature. This makes endpoints not just ends but also significant behavioral markers in the graph's context.