Problem 13 - Calculus Early Transcendentals

Question

Sketches from properties Sketch a graph of a function that is continuous on \((-\infty, \infty)\) and has the following properties. Use a sign graph to summarize information about the function. $$\begin{array}{l} f(0)=f(4)=f^{\prime}(0)=f^{\prime}(2)=f^{\prime}(4)=0 ; f(x) \geq 0 \text { on } \\ (-\infty, \infty) \end{array}$$

Step-by-Step Solution

Answer

Question: Sketch a graph of a continuous function on \((-\infty, \infty)\) with the properties \(f(0)=f(4)=0\), \(f'(0)=f'(2)=f'(4)=0\), and \(f(x)\geq0\) on \((-\infty,\infty)\). Answer: The graph of the function should consist of parabolic segments on the intervals \((-\infty,0)\), \((0,2)\), \((2,4)\), and \((4,\infty)\), with critical points at \((0,0)\), \((2,f(2))\), and \((4,0)\), where the function has a local minimum at \(x=0\) and \(x=4\) and a local maximum at \(x=2\).

Step 1: Identify the critical points and given information

We are given: 1. \(f(0)=f(4)=0\) 2. \(f'(0)=f'(2)=f'(4)=0\) 3. \(f(x)\geq0\) on \((-\infty,\infty)\). Critical points are \(x=0,2,4\), as the derivative is zero at these points. The function is non-negative.

Step 2: Set up the sign graph

Divide the domain into intervals with the critical points, i.e., \((-\infty,0)\), \((0,2)\), \((2,4)\), and \((4,\infty)\).

Step 3: Determine the sign and shape of the function in each interval

Since \(f'(0)=f'(2)=f'(4)=0\), each critical point represents a local maximum or minimum. Since \(f(x)\geq 0\), the function must have local minima at \(x=0\) and \(x=4\), and a local maximum at \(x=2\). We do not know the exact value of the local maximum since we are not given \(f(2)\), but it is clear that there is an increase between \(0\) and \(2\) and a decrease between \(2\) and \(4\). On the interval \((-\infty,0)\), the function is non-negative and must approach \(f(0)=0\). Since \(f'(0)=0\), the function should look like a parabola opening upwards. On the interval \((0,2)\), the function is increasing as it approaches a local maximum at \(x=2\). Again, the shape of the function should look like a parabola opening upwards. On the interval \((2,4)\), the function is decreasing as it approaches the local minimum \(f(4)=0\). The shape of the function should look like a parabola opening downwards. On the interval \((4,\infty)\), the function is non-negative and must approach \(f(4)=0\). Since \(f'(4)=0\), the function should look like a parabola opening upwards.

Step 4: Sketch the graph

Combining all the information from the previous steps, the graph should look like: 1. A parabola opening upwards on \((-\infty,0)\), touching the x-axis at \(x=0\) 2. A parabola opening upwards on \((0,2)\), reaching a local maximum at \(x=2\) 3. A parabola opening downwards on \((2,4)\), touching the x-axis at \(x=4\) 4. A parabola opening upwards on \((4,\infty)\) To draw the graph, start by marking the critical points \((0,0)\), \((2,f(2))\), and \((4,0)\). Then sketch the increasing parabola on \((-\infty,0)\) that touches the x-axis at \(x=0\). Next, sketch the increasing parabola on \((0,2)\) that reaches a local maximum at \(x=2\). After that, sketch the decreasing parabola on \((2,4)\) that touches the x-axis at \(x=4\). Finally, sketch the increasing parabola on \((4,\infty)\). The result should be a continuous non-negative function on \((-\infty,\infty)\) with the specified properties.

Key Concepts

Graph of a Function

When sketching the graph of a function, you visualize the behavior of the function over a set domain. Graphs provide a pictorial representation that can help us understand how a function behaves at different points, or across intervals. Here, the task is to draw a graph for a function that is continuous on \((-\infty, \infty)\) and meets specific criteria.

• Start by marking the key points provided, such as where the function and its derivative are zero.
• Pay attention to where the function increases or decreases, and where it might have peaks or troughs.
• Ensure the graph is smooth, adhering to continuity, meaning no breaks or jumps.

In this exercise, the graph must be continuously moving above or touching the x-axis since \(f(x)\geq 0\). This makes the function appear as a series of connected parabolas reflecting the character of the critical points and the intervals between them.

Critical Points

Critical points of a function occur where the first derivative equals zero or is undefined. These points often indicate where a function's graph might have local maxima, minima, or inflection points.

• For this exercise, critical points are at \(x=0,2,4\) because the derivative is zero at these points.
• Each of these points should be evaluated to determine if they are local maxima, minima, or neither.

In our scenario:

• \(x=0\) and \(x=4\) are local minima, meaning the function dips down at these points.
• \(x=2\) serves as a local maximum, representing a peak in the graph.

Identifying these critical points helps dictate the shape and smoothness of the graph and indicates the flow of the function throughout its domain.

Sign Graph

A sign graph is a helpful tool used to summarize the behavior of the derivative and, consequently, the function itself over different intervals. By mapping which sections of the function are positive, negative, or zero, we can determine the nature of the graph.

• Divide the function into intervals based on the critical points: \((-\infty,0)\), \((0,2)\), \((2,4)\), and \((4,\infty)\).
• Assess the behavior of the function within each interval by checking the sign of the first derivative.

For this exercise:

• The function displays non-negative behavior across all intervals.
• These insights help outline whether the graph will be increasing, decreasing, or constant in specific regions.

Continuity

Continuity of a function implies that it can be drawn without lifting the pencil off the paper. In mathematical terms, a function is continuous if it has no breaks, jumps, or gaps across its domain.

• To achieve continuity for \(f(x)\) over \((-\infty, \infty)\), ensure that the graph flows smoothly from one section to the next.
• This involves seamlessly moving between increases and decreases without abrupt changes in direction.

In our exercise:

• The function's continuity is evidenced by the smooth progression of the graph through the defined critical points.
• Each portion of the graph, dictated by sign changes of the derivative, connects perfectly without interruptions.

Ensuring continuity in the graph satisfies one of the primary conditions of the exercise, reflecting a flawless and predictable function behavior throughout its entire domain.