Problem 13 - Calculus Single Variable

Question

Follow the outline given in this section to give a careful sketch of the graph of each of the functions in Exercises \(1-24\). Your sketch should exhibit, and have labeled, all of the following: a) local and global extrema, b) inflection points, c) intervals on which function is increasing or decreasing, d) intervals on which function is concave up or concave down, e) horizontal and vertical asymptotes. $$ f(x)=\left(x^{1 / 3}+1\right) / x $$

Step-by-Step Solution

Answer

The critical and inflection points, asymptotes, and concavity inform the function's behavior, capturing its complete graph structure.

Step 1: Find the Critical Points

First, identify the critical points by differentiating the function \(f(x) = \frac{x^{1/3} + 1}{x}\). Apply the quotient rule: \(f'(x) = \frac{(x^{1/3} + 1)' \, x - (x^{1/3} + 1) \, 1}{x^2}\). Simplify the derivative to find critical points where \(f'(x) = 0\) or is undefined. Calculate \((x^{1/3} + 1)'\) as \(\frac{1}{3}x^{-2/3}\). This gives us the derivative: \(f'(x) = \frac{\frac{1}{3}x^{-2/3}x - (x^{1/3} + 1)}{x^2}\). Set \(f'(x) = 0\) and solve for \(x\).

Step 2: Determine Increasing/Decreasing Intervals

Using the critical points found, test intervals around them to determine where the derivative \(f'(x)\) is positive or negative. These indicate the function is increasing or decreasing, respectively. Analyze each interval by selecting a test point and substituting it into \(f'(x)\).

Step 3: Find Points of Inflection

To find inflection points, differentiate \(f'(x)\) to get \(f''(x)\). Analyze sign changes in \(f''(x)\) to identify intervals of concavity. Inflection points occur where \(f''(x)\) changes sign. Calculate \(f''(x)\) and solve for changes in sign across intervals.

Step 4: Evaluate Concavity Intervals

Using the second derivative from the previous step, evaluate the intervals to check where the function is concave up \((f''(x) > 0)\) or concave down \((f''(x) < 0)\). Determine signs using test points in the intervals.

Step 5: Identify Horizontal and Vertical Asymptotes

Horizontal asymptotes occur as \(x\) approaches infinity. Investigate \(\lim_{x \to \infty} f(x)\) and \(\lim_{x \to -\infty} f(x)\). For vertical asymptotes, find values where the denominator equals zero, so \(x = 0\) is a vertical asymptote. Generally, compute \(\lim_{x \to a} f(x)\) for undefined points.

Step 6: Mark Extrema and Intervals on Graph

Use information from previous steps to plot and label local/global extrema, inflection points, and asymptotes on a graph. Indicate increasing/decreasing and concavity intervals by clearly marking them on the sketch.

Key Concepts

Critical Points

In calculus, critical points are essential in analyzing the behavior of a function. They occur where the function's derivative is zero or undefined. For the function \(f(x) = \frac{x^{1/3} + 1}{x}\), finding critical points involves differentiating the function using the quotient rule. Applying this rule simplifies to the derivative:

• \(f'(x) = \frac{\frac{1}{3}x^{-2/3}x - (x^{1/3} + 1)}{x^2}\)

A critical point exists where this derivative equals zero or is undefined. For instance, at \(x = 0\), the function becomes undefined, suggesting a vertical asymptote rather than a critical point. By solving \(f'(x) = 0\), we find other potential critical points which are used to determine the function's behavior.

Function Behavior

Understanding a function's behavior means knowing where it increases or decreases. This requires examining the sign of its first derivative, \(f'(x)\). Once critical points are found, we test intervals of \(x\) around these points:

• If \(f'(x) > 0\), the function is increasing in that interval.
• If \(f'(x) < 0\), the function is decreasing.

Choose test points within each interval, plug them into the derivative, and observe the sign of the result. This analysis provides insights about changes in the function’s trajectory, crucial for sketching an accurate graph.

Inflection Points

Inflection points are crucial in understanding a function's concavity. A point of inflection is where the function changes its concavity, either from concave up to concave down, or vice versa.
The second derivative, \(f''(x)\), helps us find these points. Differentiating \(f'(x)\) gives \(f''(x)\). Inflection points are found where \(f''(x)\) changes sign:

• If \(f''(x) > 0\), the function is concave up.
• If \(f''(x) < 0\), the function is concave down.

Finding exact intervals of concavity requires testing intervals between potential inflection points for changes in the sign of \(f''(x)\). These points and behaviors are also marked on the graph to indicate where the curve shifts its bending direction.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. These are indicative of behavior at the extremities of the graph. There are two types of asymptotes important in this context:

• Vertical Asymptotes: These occur where the function is undefined. For \(f(x)\), \(x = 0\) leads to division by zero, creating a vertical asymptote.
• Horizontal Asymptotes: To find these, evaluate the function's limit as \(x\) approaches infinity or negative infinity. Check \(\lim_{x \to \infty} f(x)\) and \(\lim_{x \to -\infty} f(x)\).

Marking these asymptotes on a graph helps prevent skew interpretations of the function's end-behavior. They act as guidance for the direction the function heads as \(x\) becomes very large or small.