Question

Sketch the graph of a function that is continuous on \((-\infty, \infty)\) and satisfies the following sets of conditions. $$\begin{aligned} &f^{\prime \prime}(x)>0 \text { on }(-\infty,-2) ; f^{\prime \prime}(x)<0 \text { on }(-2,1) ; f^{\prime \prime}(x)>0 \text { on }\\\ &(1,3) ; f^{\prime \prime}(x)<0 \text { on }(3, \infty) \end{aligned}$$

Short answer

Question: Based on the given conditions related to the second derivative, sketch a continuous function that satisfies the concavity conditions and identify the points of inflection. Answer: The graph of the continuous function should have points of inflection at x = -2, x = 1, and x = 3, with the function being concave up in the intervals (-∞, -2) and (1, 3), and concave down in the intervals (-2, 1) and (3, ∞).

Step-by-step solution

Understand the given conditions

The given conditions are all related to the concavity of the function. We have the following information: - \(f^{\prime \prime}(x)>0\) on \((-\infty, -2)\): This means the graph of the function should be concave up in this interval. - \(f^{\prime \prime}(x)<0\) on \((-2, 1)\): The graph should be concave down in this interval. - \(f^{\prime \prime}(x)>0\) on \((1, 3)\): The graph should be concave up in this interval. - \(f^{\prime \prime}(x)<0\) on \((3, \infty)\): the graph should be concave down in this interval.

Identify the points of inflection

Since the concavity changes at \(x=-2, 1,\) and \(3\), these points must be points of inflection. The points of inflection are where the second derivative changes its sign.

Sketch the graph considering the concavity and points of inflection

Start by drawing the points of inflection at \(x=-2, 1,\) and \(3\). Then, follow the concavity conditions given: - Draw a concave-up curve starting from the leftmost part of the graph towards the point of inflection at \(x=-2\). - From the point of inflection at \(x=-2\), draw a concave-down curve to the point of inflection at \(x=1\). - From the point of inflection at \(x=1\), draw a concave-up curve to the point of inflection at \(x=3\). - Lastly, from the point of inflection at \(x=3\), draw a concave-down curve towards the rightmost part of the graph. Note that the specific shape of the graph and the position of the points of inflection in the coordinate plane are arbitrary, as long as the concavity conditions are satisfied. We sketched a graph to give a general shape that satisfies the given concavity conditions.

Key concepts

Second Derivative Test

The second derivative test is a fundamental tool in calculus for analyzing the concavity of a function and identifying possible points of inflection. By taking the second derivative of a function, denoted as \(f''(x)\), and evaluating its sign, we can gain insight into the curvature of the function's graph.

When \(f''(x) > 0\), the function is said to be concave up on that interval. This can be visualized as a section of the graph that holds water, resembling a 'smile'. On the other hand, when \(f''(x) < 0\), the function is concave down, similar to an inverted 'smile', or a region where water would pour off.

Understanding the second derivative test is crucial for sketching accurate graphs of functions, as it allows us to predict the behavior of the graph between critical points.

Sketching Graphs of Functions

Sketching graphs effectively requires combining several calculus concepts, including the analysis of the first and second derivatives. To sketch a graph, begin by determining critical points where \(f'(x) = 0\) or where \(f'(x)\) does not exist. These points help indicate where the graph might have peaks, valleys, or asymptotes.

Next, utilize the second derivative test discussed previously to understand the concavity between these points. As you sketch, pay attention to the intervals defined by the second derivative to ensure the graph reflects the correct concavity. Remember, your sketch does not have to be perfect, but it must accurately reflect the behavior of the function within the given intervals.

Understanding Concavity

Concavity is a term that describes the curvature of a function's graph. To fully grasp this concept, imagine you're holding a flexible, thin stick. If you bend the stick so it forms a 'U' shape, you've just demonstrated concave up. Flip the stick to make an 'n' shape, and it's now concave down.

Mathematically, concavity is determined by the sign of the second derivative, \(f''(x)\). Positive means concave up, and negative concave down. Understanding concavity helps in visualizing the overall shape of a graph and is especially important for various applications in physics, economics, and biology.

Identifying Points of Inflection

Points of inflection are places on a graph where the concavity changes from up to down or down to up. These points are particularly important because they signal a change in the behavior of a function's graph. To find them, look for where the second derivative \(f''(x)\) changes sign.

For instance, given the information from the exercise, the points where the second derivative changes from positive to negative (or vice versa) are potential points of inflection. These points are critical when sketching the graph as they provide a structured approach to outlining the function's curvature. It is important to note, however, that not all sign changes in the second derivative will lead to a point of inflection: the function must also be continuous at that point.