Question

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

Short answer

Question: Given a continuous function with inflection points at x = -2, x = 1, and x = 3, and concavity such that \(f^{\prime \prime}(x) > 0\) on \((-\infty, -2) \cup (1, 3)\) and \(f^{\prime \prime}(x) < 0\) on \((-2, 1) \cup (3, \infty)\), describe the graph's behavior in each interval and at the inflection points. Answer: The graph of the function behaves as follows in each interval and at the inflection points: 1. On the interval \((-\infty, -2)\), the function is concave up, starting with a negative slope and progressively becoming less steep as it approaches the inflection point \(I_1=(-2, f(-2))\). 2. On the interval \((-2, 1)\), the function is concave down, starting with a slightly positive slope at \(I_1\) and becoming steeper as it approaches the inflection point \(I_2=(1, f(1))\). 3. On the interval \((1, 3)\), the function is concave up, starting with a steep positive slope at \(I_2\) and progressively becoming less steep as it approaches the inflection point \(I_3=(3, f(3))\). 4. On the interval \((3, \infty)\), the function is concave down, starting with a slightly positive slope at \(I_3\) and becoming steeper as it extends to the right.

Step-by-step solution

Understanding the role of given conditions

We are given that: 1. \(f^{\prime \prime}(x) > 0\) on \((-\infty, -2)\). This means the function is concave up in this interval. 2. \(f^{\prime \prime}(x) < 0\) on \((-2, 1)\). This means the function is concave down in this interval. 3. \(f^{\prime \prime}(x) > 0\) on \((1, 3)\). This means the function is concave up in this interval. 4. \(f^{\prime \prime}(x) < 0\) on \((3, \infty)\). This means the function is concave down in this interval.

Identify the inflection points

Since the concavity changes at x = -2, x = 1, and x = 3, these points are the inflection points of the function. We will denote them as \(I_1=(-2, f(-2))\), \(I_2=(1, f(1))\), and \(I_3=(3, f(3))\).

Sketch the graph segment by segment

We will sketch the graph in four segments: 1. From \((-\infty, -2)\): The graph should be concave up. Starting from the left, the graph will have a negative slope and gradually become less steep as it approaches the inflection point \(I_1\). 2. From \((-2, 1)\): The graph should be concave down. From the inflection point \(I_1\), the graph should start with a slightly positive slope and become more steep as it approaches the next inflection point \(I_2\). 3. From \((1, 3)\): The graph should be concave up. From the inflection point \(I_2\), the graph should start with a steep positive slope and gradually become less steep as it approaches the inflection point \(I_3\). 4. From \((3, \infty)\): The graph should be concave down. From the inflection point \(I_3\), the graph should start with a slightly positive slope and gradually become steeper as it goes to the right.

Connect the segments

We have now sketched the graph in different intervals respecting the inflection points and concavity constraints. The last step is to make sure all segments are connected and continuous on the entire domain of the function. To summarize, the function's graph should be progressively less steep from left to right, going through the inflection points \(I_1\), \(I_2\), and \(I_3\), with concave up portions in the intervals \((-\infty, -2)\) and \((1, 3)\), and concave down portions in the intervals \((-2, 1)\) and \((3, \infty)\).

Key concepts

Concave Up

Understanding when a graph is concave up is fundamental in graph analysis. Imagine a right-side-up bowl, where any segment along the curve is similar to a portion of the bowl's edge; this is what concave up looks like. Mathematically, we determine this by evaluating the second derivative of a function, denoted as \( f''(x) \). If \( f''(x) > 0 \), the graph of the function is curving upwards. This means as you move from left to right, the slope of the tangent line to the graph is increasing.

In our exercise, intervals \( (-\text{\( \)\infty, -2}) \) and \( (1, 3) \) satisfy this condition, so the graph in these intervals will be shaped like a cup, holding up water, depicting positive concavity.

Concave Down

Conversely, a graph that is concave down will resemble an upside-down bowl. When you have \( f''(x) < 0 \), the graph dips down, creating a kind of canopy shape. This signifies that the slope of the tangent is decreasing as you proceed along the graph from left to right.

In the context of our exercise, the graph is concave down in the intervals \( (-2, 1) \) and \( (3, \text{\( \)\infty}) \), implying it arcs downwards like the top side of an umbrella.

Inflection Point

An inflection point indicates a fascinating twist in the graph's tale. It's a specific point where the graph changes its concavity — from concave up to concave down or vice versa. Think of it as a tipping point on a roller coaster where the track switches direction. It’s where the second derivative switches sign. In simple terms, it's where \( f''(x) \) goes from positive to negative, or the other way around.

Our exercise presents us with three inflection points: \( x = -2 \), \( x = 1 \), and \( x = 3 \). At these x-values, the graph changes concavity, giving it a more varied and interesting shape. Identifying these points is an essential step in sketching the entire graph.

Second Derivative Test

The Second Derivative Test is like a detective's tool for functions—it helps us find out if a point of the function is a local minimum or maximum, just based on the function's concavity. If you have a critical point (where the first derivative, \( f'(x) \), is zero or undefined) and the second derivative at that point is positive, the point is a local minimum because the graph is concave up. If the second derivative is negative, it’s a local maximum due to the concave down nature.

While the exercise does not explicitly require finding local minima or maxima, understanding how concavity shapes the graph and affects such points is crucial to visualize the overall function.

Sketching Graphs of Functions

Now, let's talk about sketching graphs of functions. Visualizing a function's graph requires blending all our knowledge of derivatives, concavity, and inflection points. Start by marking the inflection points on your x-axis. Then sketch the pieces of the graph between these points, keeping in mind whether the section should be concave up or concave down. Connect these segments smoothly, respecting continuity, and ensure that the concavity is correct throughout.

Following the step-by-step solution provided in the exercise, by determining inflection points and analyzing concavity, you can gradually build up a precise picture of how the function behaves throughout its domain, leading to a complete and accurate graph.