Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. $$ \begin{array}{l} f(0)=3 ; f(3)=0 ; f(6)=4 \\ f^{\prime}(x)<0 \text { on }(0,3) ; f^{\prime}(x)>0 \text { on }(3,6) ; \\ f^{\prime \prime}(x)>0 \text { on }(0,5) ; f^{\prime \prime}(x)<0 \text { on }(5,6) \end{array} $$

Step-by-Step Solution

Answer

Graph decreases and is concave up from \( (0, 3) \) to \( (3, 0) \), increases and remains concave up from \( (3, 0) \) to \( (5, x) \), and increases while being concave down from \( (5, x) \) to \( (6, 4) \).

Step 1: Analyze Function Values

The function passes through the points \( (0, 3) \), \( (3, 0) \), and \( (6, 4) \). This gives us three specific coordinates that must lie on the graph of the function.

Step 2: Determine Slope Behavior

The derivative \( f'(x) \) is less than zero on the interval \( (0, 3) \), meaning the slope of the function is negative and the function is decreasing from \( x = 0 \) to \( x = 3 \). Conversely, \( f'(x) \) is greater than zero on \( (3, 6) \), indicating the function increases over this interval.

Step 3: Analyze Concavity

The second derivative \( f''(x) \) is positive on \( (0, 5) \), which shows the function is concave up (shaped like a smile) on this interval. However, \( f''(x) \) is negative on \( (5, 6) \), meaning the function is concave down (shaped like a frown) there.

Step 4: Sketch the Graph Components

Starting at \( (0, 3) \), the function decreases (as a downward curve) towards \( (3, 0) \), maintaining being concave up. From \( (3, 0) \) to \( (5, x) \), it increases while remaining concave up. Finally, from \( x = 5 \) to \( x = 6 \), the function continues to increase but changes to a concave down form as it reaches \( (6, 4) \). This transition in concavity is a key visual feature of the graph.

Key Concepts

Slope Behavior

When examining the behavior of the slope of a function, we primarily focus on the first derivative, noted as \( f'(x) \). The sign of \( f'(x) \) reveals whether the function is increasing or decreasing over specific intervals. For instance, if \( f'(x) < 0 \), it means the slope is negative, and the function is decreasing. On the contrary, if \( f'(x) > 0 \), the slope is positive, indicating the function is increasing.
In our exercise, from \( x = 0 \) to \( x = 3 \), the slope \( f'(x) \) is negative, which prescribes a downward movement of the graph, starting from the point \( (0, 3) \). From \( x = 3 \) to \( x = 6 \), where \( f'(x) \) is positive, the function takes an upward path, reflecting an increase.

• A decreasing graph aligns with negative slope signs.
• An increasing graph mirrors positive slope values.

This understanding of slope behavior is essential in predicting and sketching not just the trajectory, but also the dynamic transitions of a function over its domain.

Concavity

Concavity describes how a function bends, either upwards or downwards. The second derivative, \( f''(x) \), offers insight into the concavity of a function.
When \( f''(x) > 0 \), the graph is concave up, similar to a smiley face. This situation implies that the function's rate of increase is itself increasing or, conversely, its rate of decrease is decreasing. On the other hand, \( f''(x) < 0 \) indicates a concave down graph, which resembles a frown. This means the function is slowing down its increase or speeding up its decrease.
In the given exercise, the interval \( (0, 5) \) features a concave up behavior. This scenario means the graph gently bows upwards over this entire span. Conversely, the shift to \( f''(x) < 0 \) on \( (5, 6) \) causes the graph to change to a concave down position.

• Concavity up, \( f''(x) > 0 \), enhances upward curving.
• Concavity down, \( f''(x) < 0 \), exaggerates downward curving.

Recognizing transitions in concavity, particularly at operator intervals, helps grasp the graph's full curvature tendencies.

Sketching Continuous Functions

Sketching a continuous function involves acknowledging several key properties: critical points, the nature of the slope, and concavity. With this exercise, a continuous function travels across specific points on its graph without jumps or holes, making it appear smooth and unbroken.
Begin sketching by plotting the crucial given points: \( (0, 3) \), \( (3, 0) \), and \( (6, 4) \). These form the framework of your graph. Next, use the first derivative to understand where the function decreases and increases. According to \( f'(x) \), the function descends from \( (0, 3) \) to \( (3, 0) \) and then ascends towards \( (6, 4) \).
Integrate the second derivative to exhibit the appropriate concavity: the function is concave up as it approaches \( x = 5 \) and then switches to concave down near \( x = 6 \).

• Plot the given points to outline the graph.
• Apply slope behavior to guide direction changes.
• Use concavity to adjust the curvature.

By aligning your sketch with these ordered elements of analysis, you'll generate an accurate representation of the continuous function, providing a valuable visualization for deeper understanding.