Problem 21 - Essential Calculus Early Transcendentals

Question

Sketch the graph of a function that satisfies all of the given conditions. \(f^{\prime}(x)>0\) if \(|x|<2, \quad f^{\prime}(x)<0\) if \(|x|>2\) \(f^{\prime}(-2)=0, \quad \lim _{x \rightarrow 2}\left|f^{\prime}(x)\right|=\infty, \quad f^{\prime \prime}(x)>0\) if \(x \neq 2\)

Step-by-Step Solution

Answer

The graph of \(f(x)\) has a local minimum at \(-2\), is steep at \(2\), and is concave up elsewhere.

Step 1: Understanding the Derivative Conditions

The function's first derivative, \(f'(x)\), is positive when \(|x| < 2\), meaning the function is increasing in this interval. For \(|x| > 2\), \(f'(x)\) is negative, so the function is decreasing. This suggests that \(x = 2\) is a significant point in the graph of \(f\).

Step 2: Identifying Critical Points

We know that \(f'(-2) = 0\), which indicates a critical point at \(x = -2\). Since the derivative is only zero at this point, the function could have a local maximum, minimum, or a point of inflection here. Since \(f''(x) > 0\) everywhere except at \(x = 2\), the graph changes from concave down to concave up at \(x = -2\), indicating a local minimum.

Step 3: Behavior Near x = 2

The condition \(\lim_{x \to 2}\left|f'(x)\right| = \infty\) implies that the slope of the tangent line becomes infinitely steep as \(x\) approaches 2. This suggests a vertical tangent line exists at \(x = 2\).

Step 4: Graph Concavity Information

Since \(f''(x) > 0\) for all \(x eq 2\), the graph of the function is concave up everywhere except possibly at \(x = 2\). This concavity information ensures that the graph will be 'U' shaped.

Step 5: Sketching the Graph

- Plot the critical point at \((-2, f(-2))\) as a local minimum based on the behavior of the first and second derivatives.- Draw the graph increasing from left, reaching a minimum at \(x = -2\), then continuing to increase till it steeply approaches \(x = 2\) with a vertical tangent line, indicating a sharp rise or drop until very close to \(x = 2\).- After \(x = 2\), the function decreases, indicating the graph arcs downward as \(|x| > 2\). - Ensure the entire graph remains concave up except at \(x=2\).The resulting graph should show a minimum at \(-2\), become infinitely steep at \(x = 2\), and then decrease thereafter.

Key Concepts

Derivative Conditions

In graph sketching, understanding the derivative conditions of a function is crucial. The derivative, denoted as \(f'(x)\), provides information about the function's rate of change.
When \(f'(x) > 0\), as specified for \(|x| < 2\), it indicates that the function is increasing in that interval. This tells us the slope of the tangent line to the curve is positive, and the graph rises as you move along the x-axis.
Conversely, if \(f'(x) < 0\) for \(|x| > 2\), the function decreases in that region. Here, the slope of the tangent line is negative, and the graph falls.
These conditions help determine the overall shape and direction of the graph over specific intervals. The information allows us to decide where the graph goes "uphill" or "downhill."

Critical Points

Critical points occur where the first derivative of a function is zero or undefined. These are significant because they could represent local maxima, minima, or points of inflection on the graph.
In the exercise, we have a critical point at \(x = -2\) since \(f'(-2) = 0\). This zero derivative value suggests that the function pauses its increase or decrease at this point, potentially marking an extremal point.
Since the second derivative \(f''(x) > 0\) except at \(x = 2\), it signals the graph is concave up, allowing \(x = -2\) to be a local minimum. The behavior around this critical point gives more insights into the function's turning points and curvature.

Concavity

Concavity describes how a function curves. It is determined by the sign of the second derivative \(f''(x)\).
When \(f''(x) > 0\), the graph of the function is concave up, resembling a "U" shape. This means the rate of increase of the function is gettingstronger, or a decrease is slowing down. Visually, this forms a bowl-like structure.
In this exercise, except at \(x = 2\), \(f''(x) > 0\) everywhere else, confirming that the graph is mostly concave up. It tells us to expect a smooth, upwardcurve in large sections of the graph.

Vertical Tangent Line

A vertical tangent line occurs when the derivative of a function approaches infinity as \(x\) approaches a point.
In the given exercise, the limit condition \(\lim_{x \to 2}\left|f'(x)\right| = \infty\) suggests the slope becomes infinitely steep at \(x = 2\).
This sharp increase or decrease means at \(x = 2\), the graph has a vertical tangent, forming a sharp transition in the direction of the curve. It is an essential featureas it clearly indicates a point of major change without crossing into other sections of the graph directly.
Understanding where vertical tangents occur helps in creating an accurate and visually complete graph. Such points significantly influence theshape and movement of the graph.