Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of a function \(f\) that has the following properties: (a) \(f\) is everywhere continuous; (b) \(f(2)=-3, f(6)=1\); (c) \(f^{\prime}(2)=0, f^{\prime}(x)>0\) for \(x \neq 2, f^{\prime}(6)=3\); (d) \(f^{\prime \prime}(6)=0, f^{\prime \prime}(x)>0\) for \(26\).

Step-by-Step Solution

Answer

Sketch starts at (2,-3), is initially concave up, becomes concave down at (6,1), and is always increasing.

Step 1: Understand the Continuity Requirement

The function is everywhere continuous, which means there are no breaks, jumps, or holes in the graph of the function. We should be able to draw it without lifting the pencil from the paper.

Step 2: Plot Given Points

Mark the points on the graph corresponding to the values given: \( (2, -3) \) and \( (6, 1) \). These are points through which the graph must pass.

Step 3: Analyze and Use First Derivative Conditions

The derivative \( f'(2) = 0 \) indicates a horizontal tangent at \( x = 2 \). Since \( f'(x) > 0 \) for all \( x eq 2 \), except possibly near \( x = 2 \), the function is increasing for other values of \( x \). At \( x = 6 \), \( f'(6) = 3 \), further confirming the function is increasing at this point.

Step 4: Analyze and Use Second Derivative Conditions

\( f''(6) = 0 \) implies the presence of an inflection point at \( x = 6 \). From \( 2 < x < 6 \), \( f''(x) > 0 \) means the graph is concave up. For \( x > 6 \), \( f''(x) < 0 \) indicates concave down behavior.

Step 5: Sketch the Graph

Start at \( (2, -3) \) with a horizontal tangent, rising towards \( (6, 1) \) as a smooth concave up curve. After reaching \( (6, 1) \), inflect to concave down as the graph continues to rise. Ensure the segments of the graph transition smoothly, maintaining continuity.

Key Concepts

Graph Sketching

Graph sketching is a key part of visualizing mathematical functions. It involves creating a picture of a function based on given properties.
To effectively sketch a graph, it's important to consider several features:

• Continuity: This ensures the graph can be drawn without lifting your pencil.
• Points: Specific coordinates where the function passes.
• Shape: Understanding of concavity and slope to depict how the graph curves.
• Behavior: How the graph looks and adjusts at different sections based on changes in derivatives.

When sketching, start by plotting given points. Then, consider increasing or decreasing trends indicated by the first derivative (slope). Finally, include curves and changes in direction as dictated by the second derivative.

Derivative

The derivative of a function is a measure of its rate of change at any given point. Understanding derivatives is crucial for graph sketching because they inform us about the slope of the graph.
The first derivative, denoted as \( f'(x) \), will tell if a function is increasing or decreasing:

• If \( f'(x) > 0 \), the function is rising.
• If \( f'(x) < 0 \), the function is falling.
• If \( f'(x) = 0 \), there is a horizontal tangent, indicating a potential maximum, minimum, or inflection point.

In this exercise, \( f'(2) = 0 \) indicates a horizontal tangent at \( x = 2 \), while \( f'(6) = 3 \) suggests the function is still increasing at \( x = 6 \). Studying derivatives allows us to predict and visualize the slope and direction of the graph.

Critical Points

Critical points are important features of a function's graph where the derivative is zero or undefined. These points can signal potential maximums, minimums, or plateau regions within the graph.
In our context:

• A horizontal tangent at \( x = 2 \) suggests a critical point. This means the slope at this point is zero, implying a stationary point where the graph levels out momentarily.

To determine whether these critical points are maxima, minima, or neither, additional testing such as the second derivative test can be used. This involves examining the sign of the second derivative at the critical point: if \( f''(x) > 0 \), it's a local minimum, if \( f''(x) < 0 \), it's a local maximum.

Inflection Points

Inflection points are where the graph of a function changes concavity. Understanding inflection points helps in sketching accurate curves in the right direction.
For our function, the second derivative \( f''(x) \) tells us about the concavity of the graph:

• If \( f''(x) > 0 \), the graph is concave up (U-shaped).
• If \( f''(x) < 0 \), the graph is concave down (n-shaped).
• If \( f''(x) = 0 \), there is a possible inflection point where the concavity changes.

At \( x = 6 \), \( f''(6) = 0 \) confirms an inflection point. Before this point, the function is concave up, and after this point, it switches to concave down. Identifying inflection points helps in sketching the correct curvature in specific sections of the graph.