Problem 21 - 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. $$ f(x)=\sqrt{x} /(x+1) $$

Step-by-Step Solution

Answer

Domain: \( x \geq 0 \). Local and global extrema from critical points. Inflection points and intervals of concavity from \( f''(x) \). Horizontal asymptote at \( y=0 \).

Step 1: Identify Domain and Range

The function given is \( f(x) = \frac{\sqrt{x}}{x+1} \). For this function to be defined, the expression under the square root, \( x \), must be non-negative. Additionally, since it's a rational function, \( x + 1 \) cannot be zero. Thus, the domain of the function is \( x \geq 0 \) and \( x eq -1 \), but since \( x = -1 \) is not in \( x \geq 0 \), the effective domain is \( x \geq 0 \).

Step 2: Find Extrema

To find local extrema, we need to find the critical points by taking the derivative of \( f(x) \) and setting it equal to zero. Using the quotient rule, \( f'(x) = \frac{(x+1)(1/2x^{-1/2}) - \sqrt{x}(1)}{(x+1)^2} \). Setting \( f'(x) = 0 \) and solving, we identify critical points where extrema can occur.

Step 3: Determine Increasing/Decreasing Intervals

Evaluate the sign of \( f'(x) \) from Step 2. If \( f'(x) > 0 \), then the function is increasing, and if \( f'(x) < 0 \), it is decreasing. Solve \( f'(x) = 0 \) to find where changes occur and use test points in each interval to determine the sign.

Step 4: Find Inflection Points

To find inflection points, we need the second derivative, \( f''(x) \). Compute \( f''(x) \) of the simplified version of the first derivative and determine where \( f''(x) = 0 \) or is undefined, and check these points for changes in concavity.

Step 5: Determine Concavity Intervals

Using \( f''(x) \), test intervals around the solutions to \( f''(x) = 0 \) to decide the intervals of concavity: if \( f''(x) > 0 \), the graph is concave up; if \( f''(x) < 0 \), it is concave down.

Step 6: Identify Asymptotes

Since \( x+1 = 0 \) when \( x = -1 \), and this value is not in the domain, it does not contribute a vertical asymptote. As \( x \to \infty \), \( f(x) \to 0 \), suggesting a horizontal asymptote at \( y = 0 \).

Step 7: Sketch and Label

Sketch the graph using findings: plot critical points (extrema), transition points (inflection points), and asymptote location. Label each part with the appropriate feature.

Key Concepts

Local and Global Extrema

Local and global extrema refer to the maximum and minimum values a function can take. Local extrema occur within a specific interval, while global extrema are the highest or lowest points over the entire domain of the function. To find these extrema, one must first derive the function, which involves finding its first derivative.

Critical points are values of \( x \) where the derivative is zero or undefined. These points help identify possible locations for extrema.
For the function \( f(x) = \frac{\sqrt{x}}{x+1} \), using the quotient rule gives us the derivative, \( f'(x) \). Setting this derivative to zero helps find the critical points. Once these points are identified, analyze surrounding points to determine if they result in maxima (where the slope changes from positive to negative) or minima (from negative to positive).

Utilizing this method ensures you capture where the function reaches its high and low points relative to its neighbors.

Inflection Points

Inflection points are where the function changes its concavity, typically involving a change of the second derivative's sign. This concept highlights transitions in the graph's curvature, indicating a shift from concave up to concave down or vice versa.

For our function \( f(x) = \frac{\sqrt{x}}{x+1} \), evaluate the second derivative \( f''(x) \) which follows from differentiating \( f'(x) \). Statements like \( f''(x) = 0 \) or where it's undefined can mark inflection points. Analyze these points to see if concavity changes—this could be indicated by a sign change in \( f''(x) \).

Discovering inflection points helps clarify where the graph's shape alters, providing more insight into the function's behavior across its range.

Asymptotes

Understanding asymptotes is crucial when sketching graphs, as they represent boundaries the function approaches but does not cross. There are two types of asymptotes: horizontal and vertical.

A horizontal asymptote occurs when the function approaches a particular value as \( x \) approaches infinity. In this function, as \( x \to \infty \), \( f(x) \to 0 \), indicating a horizontal asymptote at \( y = 0 \).

Vertical asymptotes occur where a function becomes unbounded as \( x \) approaches a particular value. For \( f(x) \), due to the root and denominator, \( x = -1 \) is not in the domain, so no vertical asymptotes arise from this. Understanding these asymptotes helps position and sketch the graph accurately.

Concavity

Concavity describes whether a graph curves upward or downward, akin to a bowl's opening. To identify this, check the second derivative, \( f''(x) \).

If \( f''(x) > 0 \), the function is concave up, meaning the graph arches upward similarly to the inside of a bowl. If \( f''(x) < 0 \), it is concave down, like an upside-down bowl. Concavity tells us how a function's slope is changing and provides context for interpreting its graph.

Finding where \( f''(x) \) changes sign highlights the function's concavity transitions, aligning with the presence of inflection points, and helps in sketching a comprehensive graph that accurately reflects the function's behavior.

Increasing and Decreasing Intervals

Increasing and decreasing intervals are where the function's graph either moves upward or downward, respectively. Use the first derivative to identify these segments.

Analyze \( f'(x) \) for signs; if \( f'(x) > 0 \), the function is increasing. Conversely, \( f'(x) < 0 \) means it's decreasing. Knowing where these transitions happen offers clearer graph depiction insights.
Consider test points around critical points and solve \( f'(x) = 0 \) to assess intervals properly.

By understanding these intervals, you can better plot the function, underscoring how it behaves between turning points and through different segments of its domain.