Problem 37 - Calculus Single Variable

Question

Sketch the graph of each of the functions in Exercise \(25-40,\) exhibiting and labeling: a) all local and globa extrema; b) inflection points; c) intervals on which the func tion is increasing or decreasing; d) intervals on which the function is concave up or concave down; e) all horizontal an vertical asymptotes. $$ f(x)=|x| \cdot(x+1)^{2} $$

Step-by-Step Solution

Answer

The function is defined everywhere, increasing for \(x > 0\), and decreasing for \(x < 0\). It has no asymmetric features and changes concavity at determined points.

Step 1: Identify Domain

The function \( f(x) = |x| \cdot (x+1)^2 \) is defined for all real numbers since both \(|x|\) and \((x+1)^2\) are defined everywhere. Thus, the domain is \(-\infty, \infty\).

Step 2: Find Critical Points

To find the critical points, first compute the derivative of the function. The derivative is determined using the product rule and chain rule: \( f'(x) = \frac{d}{dx} [|x|(x+1)^2] = |x| \cdot 2(x+1) + \frac{x}{|x|}(x+1)^2 \). Simplify to find the points where \(f'(x) = 0\) or does not exist. The derivative is zero or undefined when \(x = 0\) or the expression simplifies to certain values.

Step 3: Identify Intervals of Increasing/Decreasing Behavior

Use the critical points found in Step 2 to determine intervals. Test intervals around these points using the first derivative: - If \(f'(x) > 0\), \(f(x)\) is increasing there.- If \(f'(x) < 0\), \(f(x)\) is decreasing there.

Step 4: Find Concavity and Inflection Points

Compute the second derivative, \(f''(x)\), to examine concavity: \( f''(x) = \frac{d}{dx} [f'(x)] \). Determine where \(f''(x) = 0\) or changes sign, as these points indicate possible inflection points and concavity (concave up/concdave down) behavior.

Step 5: Evaluate Horizontal Asymptotes

Investigate the behavior of \(f(x)\) as \(x \to \infty\) and \(x \to -\infty\).Since \(|x|(x+1)^2\) grows indefinitely in both positive and negative directions, there are no horizontal asymptotes.

Step 6: Check Vertical Asymptotes

Determine if there are any undefined points in \(f(x)\) that might cause vertical asymptotes. For \(f(x) = |x|(x+1)^2\), there are no real number values that make the function undefined, hence no vertical asymptotes.

Step 7: Graph the Function

Utilize the results from previous steps: - Plot critical points and inflection points. - Mask transitions from increasing to decreasing (and vice versa) behavior. - Illustrate regions of concavity up and down. Ensure features such as the lack of asymptotes are clearly shown.

Key Concepts

Local Extrema

Local extrema are the points where a function changes direction from increasing to decreasing or vice versa. For the function \( f(x) = |x| \cdot (x+1)^2 \), these are found by determining where its derivative \( f'(x) \) is zero or does not exist.
Critical points are established when we solve \( f'(x) = 0 \). Here, it is found that the critical point occurs at \( x = 0 \).
At this point, you examine the derivative by testing intervals around \( x = 0 \):

• If the sign changes from positive to negative, then it's a local maximum.
• If it changes from negative to positive, it's a local minimum.

In this case, checking these tests reveals that \( x = 0 \) represents a local minimum of the function.
Local extrema help us understand the turning points of the function, indicating where the function reaches peaks or valleys within a certain interval.

Inflection Points

Inflection points occur where the function changes concavity from concave up to concave down, or vice versa. To find these points for the function \( f(x) = |x|(x+1)^2 \), we must examine the second derivative \( f''(x) \).
The second derivative provides information on how the slope changes, which is essential for identifying concavity changes. An inflection point is discovered when \( f''(x) = 0 \) and the sign of \( f''(x) \) changes around this point.
Checking \( f''(x) = 0 \) and analyzing the intervals around these values are crucial steps:

• If \( f''(x) \) changes from positive to negative, the function changes from concave up to concave down.
• Conversely, if \( f''(x) \) changes from negative to positive, it goes from concave down to concave up.

Inflection points are integral for sketching because they show where the curvature of the graph changes, playing a significant role in its overall shape.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually reaches. In the function \( f(x) = |x|(x+1)^2 \), these are determined by examining both the limits as \( x \to \infty \) and \( x \to -\infty \), as well as undefined points.
To determine horizontal asymptotes:

• Evaluate \( \lim_{x \to \pm\infty} f(x) \).
• If this limit approaches a constant, the function has a horizontal asymptote.

In our function, \( |x|(x+1)^2 \) grows indefinitely in both directions. Therefore, no horizontal asymptotes exist.
Vertical asymptotes occur where a function becomes undefined. Here, since all parts of the function \( |x|(x+1)^2 \) are always defined for every real number, there are no vertical asymptotes.
Understanding asymptotes is crucial as they describe the behavior at the edges of the graph, influencing its approach towards infinity.

Concavity

Concavity refers to the "bending" nature of graphs, describing whether they open upwards or downwards. Determining the concavity involves the second derivative \( f''(x) \):

• If \( f''(x) > 0 \), the function is concave up (looks like a valley).
• If \( f''(x) < 0 \), it is concave down (resembles a hill).

For \( f(x) = |x|(x+1)^2 \), you need to compute \( f''(x) \) and look for intervals where the concavity changes.
These factors contribute massively to the construction of the function's graph, giving us insight into how sharply or smoothly the function curves over its domain.
This concept is exceedingly helpful when trying to visualize or sketch the graph, as it tells us how steep or gentle the upward or downward motion is.

Critical Points

Critical points are where a function's derivative is zero or undefined and indicate potential local maxima or minima and points of inflection. For the function \( f(x) = |x|(x+1)^2 \), finding \( f'(x) \) is essential to locating these points.
The derivative \( f'(x) \) becomes

• 0 where \( |x| \cdot 2(x+1) + \frac{x}{|x|}(x+1)^2 = 0 \).

For \( x = 0 \), \( f'(x) = 0 \), establishing a critical point here. Testing surrounding intervals shows where the function increases or decreases.
Identifying critical points such as this gives us valuable insights into the function's behavior, helping us predict and draw accurate graphical representations.
In summary, critical points are indispensable for understanding where functions take their extreme values and change direction.