Problem 20 - Essential Calculus Early Transcendentals

Question

Sketch the graph of a function that satisfies all of the given conditions. \(f^{\prime}(1)=f^{\prime}(-1)=0, \quad f^{\prime}(x)<0\) if \(|x|<1\) \(f^{\prime}(x)>0\) if \(1<|x|<2, \quad f^{\prime}(x)=-1\) if \(|x|>2\) \(f^{\prime \prime}(x)<0\) if \(-2 < x<0, \quad\) inflection point \((0,1)\)

Step-by-Step Solution

Answer

The graph decreases on \((-1,1)\), increases on \(1<|x|<2\), and decreases for \(|x| > 2\), with a concave down curve on \(-2 < x < 0\) and an inflection point at \((0,1)\).

Step 1: Analyze Critical Points and Intervals

First, identify where the derivative is zero: \(f^{\prime}(1) = 0\) and \(f^{\prime}(-1) = 0\). These points are potential local maxima or minima. Next, consider the intervals based on the conditions given. Since \(f^{\prime}(x) < 0\) for \(|x| < 1\), the function is decreasing in \((-1,1)\). At \(1<|x|<2\), \(f^{\prime}(x) > 0\), so the function is increasing. Lastly, for \(|x| > 2\), \(f^{\prime}(x) = -1\), indicating a constant negative slope.

Step 2: Determine Concavity and Inflection

Examine the concavity conditions: \(f^{\prime \prime}(x) < 0\) for \(-2 < x < 0\), meaning the function is concave down in this interval. The problem states there is an inflection point at \((0,1)\), where the concavity changes.

Step 3: Sketch the Graph

Start at \((x > 2)\), where the slope is a constant \(-1\), so the graph will have a straight line with a downward slope. As \(x\) approaches 2 from the left, the function increases, meeting a peak at \(x = 1\). Between \(-1\) and \(1\), it decreases to a local minimum at \(x = -1\). On \((-2, -1)\), the function is concave down, meets the inflection point at \((0,1)\) then increases towards another peak at \(x = -2\).

Key Concepts

Critical Points

Every function has spots where its derivative equals zero, called critical points. These points are important because they signal locations where the function’s slope is flat. Think of it as a car pausing to decide whether to roll uphill or downhill. In our exercise, we have critical points at \(x = 1\) and \(x = -1\). Since the derivative \(f'(x)\) at these points is zero, these are the places where the function might have local maxima or minima.
In simpler terms, the graph either takes a breath at these points before going on a climb if it’s at a minimum or starts its descent if it’s at a maximum. But just identifying critical points isn't enough; we need to combine them with derivative signs around them to see the full picture. That brings us to the next part: intervals and their behavior.

Concavity

Concavity tells us how the curve of a graph opens. Imagine if you are pouring water into a curve. If it can hold water, the curve is concave up, similar to a cup. But if it spills the water, it's concave down, much like an umbrella.
For our function, it’s specified that between \(-2 < x < 0\), we have \(f''(x) < 0\). This means the graph curves downwards, forming a frown, and water would spill off it if turned into a cup shape. Concavity gives insight into how the function is bending and helps to predict any possible inflection points, where this bending nature might change. This change of bend is precisely what we'll tackle next.

Inflection Points

An inflection point is where a graph changes its curve’s bending direction. It’s like the place on a rollercoaster where you move from a hill to a dip. It marks a point where the curvature nature switches—for example, from concave down to concave up or vice versa.
In this problem, we are informed about an inflection point at \((0,1)\). This means precisely at \(x = 0\), the graph goes from bending one way to the other. Recognizing an inflection point helps to understand the graph's visual behavior at that spot, ensuring the graph's portrayal matches the provided curvature data. Identifying these points is crucial for an accurate graph since they contribute to the graph’s overall shape.

Derivative Analysis

A derivative analysis helps us understand how a function is changing. By examining \(f'(x)\) and its expressions across different ranges, we can decipher whether the function is increasing, decreasing, or remaining constant.
Consider the given conditions: for \(|x|<1\), \(f'(x)<0\), meaning the function is decreasing. Between \(1<|x|<2\), \(f'(x)>0\) tells us it's on the rise. Beyond these, \(f'(x)=-1\) for \(|x|>2\) translates to the function having a uniform negative slope, like a steady descent. This layout of change is the roadmap to sketch an accurate graph.
To visualize these interpretations, imagine them as different chapters in the function's story—noting changes in direction, speed, and breaks. It sets the foundation to satisfactorily sketch and understand the function’s progression over its domain.