Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. $$ \begin{array}{l} f(0)=f(3)=3 ; f(2)=4 ; f(4)=2 ; f(6)=0 \\ f^{\prime}(x)>0 \text { on }(0,2) ; f^{\prime}(x)<0 \text { on }(2,4) \cup(4,5) ; \\ f^{\prime}(2)=f^{\prime}(4)=0 ; f^{\prime}(x)=-1 \text { on }(5,6) \\ f^{\prime \prime}(x)<0 \text { on }(0,3) \cup(4,5) ; f^{\prime \prime}(x)>0 \text { on }(3,4) \end{array} $$

Step-by-Step Solution

Answer

Plot key points, analyze derivatives, and sketch a graph meeting all conditions.

Step 1: Plot Key Points

Identify and plot the given key points on the function. These points are \((0, 3)\), \((3, 3)\), \((2, 4)\), \((4, 2)\), and \((6, 0)\). These will serve as important locations where the function changes behavior.

Step 2: Analyze Derivative Conditions

Examine the behavior of \(f'(x)\). It is given that \(f'(x) > 0\) on \((0,2)\), meaning the function is increasing up to \(x = 2\). Then, \(f'(x) < 0\) on \((2,4)\) and \((4,5)\), indicating the function decreases between 2 and 4, and continues to decrease from 4 to 5. Specifically, \(f'(2) = 0\) and \(f'(4) = 0\) show local extrema at these points. Finally, \(f'(x) = -1\) on \((5,6)\), meaning a constant negative slope in this region.

Step 3: Analyze Concavity with Second Derivative

Consider \(f''(x)\). The function is given as concave down on \((0,3)\) and \((4,5)\), meaning it curves downward in these intervals. Conversely, it is concave up on \((3,4)\), indicating an upward curving in this interval.

Step 4: Sketch the Graph

Using the information from the previous steps, start sketching: - From \((0, 3)\) to \((2, 4)\), draw an increasing curve reaching a peak at \((2,4)\), this curve should be concave down. - From \((2, 4)\) to \((3, 3)\), draw a decreasing curve that begins to change concavity from downward to upward. - Between \((3, 3)\) and \((4, 2)\), draw a decreasing curve that is concave upward. - From \((4, 2)\) to \((5, 1)\), draw a decreasing curve that is concave downward. - Finally, from \((5, 1)\) to \((6, 0)\), draw a linear decline with slope \(-1\).

Step 5: Review and Verify

Review the sketch to ensure it meets all conditions: the plotted points match given coordinates, the slopes change as described by the first derivative conditions, and the curve follows the concavity described by the second derivative conditions. Confirm that the graph is continuous throughout \([0, 6]\) without jumps or breaks.

Key Concepts

Continuous Function

A continuous function is one with no breaks, jumps, or holes in its graph over a specific interval. For this exercise, the function is defined on \( [0, 6] \), meaning it is smooth and uninterrupted across this range.
To sketch a continuous function, we focus on key points (given points through which the function passes) and ensure that the function smoothly connects these points. In this example, the function goes through \( (0, 3), (3, 3), (2, 4), (4, 2),\) and \( (6, 0)\).
The overall goal is to create a graph that smoothly transitions through each given point without any discontinuity. This means when moving from left to right, the pencil shouldn’t leave the paper. It’s crucial to confirm that all changes in direction and slope are within the given constraints to maintain continuity.

Derivative Analysis

Derivative analysis provides insight into how a function behaves in terms of increasing or decreasing, and finding extremas like peaks or troughs. By looking at \( f'(x) \), we can determine where the function rises or falls.

• When \( f'(x) > 0 \), the function increases, as seen from \( (0, 2)\) in this exercise.
• At points where \( f'(x) = 0 \), the function has potential local maxima or minima, such as at \( x = 2 \) and \( x = 4 \).
• Where \( f'(x) < 0 \,\) the function decreases; this behavior is noted between \( (2, 4),\) \( (4, 5),\) and \((5, 6)\).

By studying these conditions, we can accurately depict how the function behaves between key points and locate any important features where the behavior changes.

Concavity

Concavity refers to how a curve bends or the direction it curves. It is characterized by the second derivative, \( f''(x) \).

• If \( f''(x) < 0 \), the function is concave down, leading the graph to curve downward like a frown. In this exercise, this occurs from \( (0, 3)\) and \( (4, 5)\).
• If \( f''(x) > 0 \), the function is concave up, making the graph curve upward like a smile, observed between \( (3, 4)\).

Understanding concavity helps in visualizing the curvature of the graph. Since changes in concavity often accompany inflection points, analyzing these helps provide a complete picture of the graph’s form. This aids in creating an accurate sketch that adheres to the prescribed curve behaviors.

Problem-Solving in Calculus

Problem-solving in calculus involves combining various analytical methods to accurately depict mathematical situations. By using calculus, one can predict how a function's graph should look based on given conditions.
Here’s how these elements work together:

• Identify important points along the graph, which are input for plotting and guiding the graph’s shape.
• Analyze derivatives to understand changes in direction and to locate extrema.
• Use concavity information to predict the curvature between key points.
• Finally, integrate all information to sketch a graph fulfilling all conditions such as continuity and slope behavior.

Successful calculus problem-solving requires comprehending each component and executing them precisely to depict real-world situations or purely theoretical functions as in this exercise.