Question

Sketch the graph of a function that is continuous on \((-\infty, \infty)\) and satisfies the following sets of conditions. $$\begin{array}{l} f(-2)=f^{\prime \prime}(-1)=0 ; f^{\prime}\left(-\frac{3}{2}\right)=0 ; f(0)=f^{\prime}(0)=0 \\ f(1)=f^{\prime}(1)=0 \end{array}$$

Short answer

Question: Sketch the graph of a function that is continuous on \((-\infty, \infty)\) and satisfies the following conditions: \(f(-2) = f^{\prime\prime}(-1) = 0\), \(f^{\prime}\left(-\frac{3}{2}\right) = 0\), \(f(0) = f^{\prime}(0) = 0\), and \(f(1) = f^{\prime}(1) = 0\). Answer: 1. Identify critical points: \((-2, 0)\), \(\left(-\frac{3}{2}, f\left(-\frac{3}{2}\right)\right)\), \((0, 0)\), and \((1, 0)\). 2. Analyze derivatives: Changes in concavity at \(x=-1\) and local extrema at \(x=-\frac{3}{2}\), \(x=0\), and \(x=1\). 3. Determine function's behavior: Local extremum between \((-2, 0)\) and \((-\frac{3}{2}, f\left(-\frac{3}{2}\right))\); concavity change at \(x=-1\); local extrema at \(x=0\) and \(x=1\). 4. Sketch graph: Plot critical points, draw graph considering function's behavior, and connect the graph ensuring continuity. Note that the sketch is an approximation, and the exact function that satisfies these conditions may not be unique. However, by following these steps, you have a rough sketch that captures the key features of the function.

Step-by-step solution

Identify Critical Points

Based on the given conditions, we can identify the following critical points: \((-2, 0)\), \(\left(-\frac{3}{2}, f\left(-\frac{3}{2}\right)\right)\), \((0, 0)\), and \((1, 0)\). These points will be the basis of our graph.

Analyze Derivatives

In order to sketch the graph, we need to use the given conditions on the first and second derivatives to understand the function's behavior around the critical points: 1. \(f^{\prime\prime}(-1) = 0\) - This indicates that the function changes concavity at \(x=-1\). 2. \(f^{\prime}\left(-\frac{3}{2}\right) = 0\) - This indicates that there is a local extremum at \(x=-\frac{3}{2}\). 3. \(f^{\prime}(0) = 0\) - This indicates that there is another local extremum at \(x=0\). 4. \(f^{\prime}(1) = 0\) - This indicates that there is another local extremum at \(x=1\).

Determine the Function's Behavior

Now, we can determine the behavior of the function around the critical points based on the information we have: - Between \((-2, 0)\) and \((-\frac{3}{2}, f\left(-\frac{3}{2}\right))\), there is a local extremum. - At \(x = -1\), the function changes concavity. - Between \((0, 0)\) and \((1, 0)\), there is a local extremum at both points

Sketch the Graph

Based on the above information, we can now roughly sketch the graph. Remember that this sketch is not unique, as there are infinite functions that may satisfy these conditions: 1. Plot the critical points: \((-2, 0)\), \(\left(-\frac{3}{2}, f\left(-\frac{3}{2}\right)\right)\), \((0, 0)\), and \((1, 0)\). 2. Draw the graph going through these points, keeping in mind the function's behavior around them. 3. Make sure that between \((-2, 0)\) and \((-\frac{3}{2}, f\left(-\frac{3}{2}\right))\) there is a local extremum and that the function changes concavity at \(x = -1\). 4. Also, ensure that there are local extrema at \(x=0\) and \(x=1\). 5. Finally, connect the graph ensuring it is continuous everywhere. Remember that the sketch is an approximation of the graph, and the exact function that satisfies these conditions is not unique. However, by following these steps, you should have a rough sketch that captures the key features of the function.

Key concepts

Critical Points in Calculus

Critical points in calculus are places on the graph of a function where its derivative is either zero or undefined. These points are important because they can indicate where the function has a local maximum or minimum -- in other words, where the graph peaks or troughs.

From the exercise conditions, \(f^{\text{prime}}(-\frac{3}{2}) = 0\), \(f^{\text{prime}}(0) = 0\), and \(f^{\text{prime}}(1) = 0\), we can deduce that \(x = -\frac{3}{2}\), \(x = 0\), and \(x = 1\) are critical points. But identifying critical points is just the first step. To truly understand the behavior of the function at these points, we need to study the sign of the derivative before and after each critical point to know whether they represent a local maximum, minimum, or a saddle point (where the derivative changes sign).

Application in Sketching Graphs: When graphing a function, marking the critical points helps us plot where the graph will change direction. This allows students to sketch more accurate graphs, especially highlighting where the function reaches its relative highs and lows.

In our example, we would plot the critical points on the graph and check the behavior of the function at those points to determine if they are maxima, minima, or neither. This helps in creating a more precise sketch of the original function.

Behavior of Derivatives

The behavior of derivatives is essential in understanding how a function is changing. The first derivative \(f^{\text{prime}}\) tells us about the rate of change of the function -- whether the function is increasing or decreasing, and how steeply. When the first derivative is positive, the function is increasing; when it is negative, the function is decreasing.

The second derivative \(f^{\text{prime prime}}\), on the other hand, gives us information about the concavity of the function. If \(f^{\text{prime prime}}\) is positive, the graph is concave up, resembling a U-shaped curve, and if it's negative, the graph is concave down, like an upside-down U.

In the given exercise, \(f^{\text{prime prime}}(-1) = 0\) tells us that there's a change in concavity at \(x = -1\). This might represent an inflection point where the graph changes from concave up to concave down, or vice versa. Understanding how the derivative behaves around this point provides insight into how the graph transitions between different segments. Knowing this change helps anticipate the shape of the graph when we sketch it, resulting in a more accurate representation.

Concavity and Inflection Points

Concavity refers to the curvature of a graph of a function. If a graph is concave up, like the interior of a cup, the function is increasing at an increasing rate. Conversely, if a graph is concave down, like an arch, the function is increasing at a decreasing rate or decreasing at an increasing rate.

Inflection points are where the graph of a function changes concavity, from concave up to concave down, or vice versa. They are significant as they signal a change in the behavior of the function's rate of change. For inflection points to occur, the second derivative of the function \(f^{\text{prime prime}}\) typically has to change sign.

In the problem, we see that \(f^{\text{prime prime}}(-1) = 0\) which suggests a potential inflection point at \(x = -1\). When sketching functions, inflection points are important as they're pivotal in the function's graph. They don't necessarily equate to local maxima or minima, but they do indicate where the shape of the graph makes a distinct change.

By identifying where these inflection points occur, students can produce more accurate, nuanced sketches of functions. These points contribute substantially to understanding the overall geometry and behavior of the function's graph in the greater context of the coordinate plane.