Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). $$ g(x)=\frac{\ln (x+1)}{x+1} $$

Step-by-Step Solution

Answer

The function is increasing for \( x > e^2 - 1 \) and decreasing before. It's concave up before \( x = e^2 - 1 \) and concave down after.

Step 1: Find the Domain

The function \( g(x) = \frac{\ln(x+1)}{x+1} \) is defined for values where \( x+1 > 0 \). Therefore, the domain of \( g(x) \) is \( x > -1 \).

Step 2: Find the First Derivative

Use the quotient rule to find the first derivative \( g'(x) \). The quotient rule is \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \), with \( u = \ln(x+1) \) and \( v = x+1 \). This gives: \[ g'(x) = \frac{(1)(x+1) - \ln(x+1)(1)}{(x+1)^2} = \frac{x+1 - \ln(x+1)}{(x+1)^2} \]

Step 3: Determine Where the Function is Increasing or Decreasing

Set \( g'(x) = 0 \) to find critical points. Solve \( x+1 - \ln(x+1) = 0 \) for \( x \). Analyze the sign of \( g'(x) \) by testing intervals around your critical points to find where the function is increasing (\( g'(x) > 0 \)) and decreasing (\( g'(x) < 0 \)).

Step 4: Find the Second Derivative

Find the second derivative \( g''(x) \) using the quotient rule on \( g'(x) \):\[ g''(x) = \frac{(x+1)(1/(x+1)) - (1)(x+1-\ln(x+1))}{(x+1)^4} \] Simplify to determine concavity:\[ g''(x) = \frac{\ln(x+1) - 2}{(x+1)^4} \]

Step 5: Determine Concavity and Points of Inflection

Set \( g''(x) = 0 \) to find possible points of inflection: \( \ln(x+1) - 2 = 0 \Rightarrow x+1 = e^2 \Rightarrow x = e^2 - 1 \). Test intervals between critical points found to determine where the graph is concave up (\( g''(x) > 0 \)) and concave down (\( g''(x) < 0 \)).

Step 6: Sketch the Graph

Using information about increasing, decreasing, concave up, and concave down intervals, sketch a smooth curve representing \( g(x) \). Indicate key points such as the critical points and points of inflection, and ensure that the graph aligns with the identified behaviors in steps 3 and 5.

Key Concepts

Derivative

In calculus, the derivative is a powerful tool we use to understand the behavior of functions. Essentially, the derivative at a point tells us the slope of the tangent line to the graph of a function at that particular point. For the function \( g(x) = \frac{\ln (x+1)}{x+1} \), to find the first derivative \( g'(x) \), we apply the quotient rule.

• The first derivative provides vital information about rates of change and trends such as where the function is increasing or decreasing.
• Using the quotient rule, \( g'(x) \) helps us determine the slopes at various points on the curve.

By setting \( g'(x) = 0 \), we can find critical points which are candidates where local maxima or minima might occur. Analyzing the sign of \( g'(x) \) around these critical points shows us where the function increases or decreases.

Concavity

The concept of concavity delves deeper into the function's behavior by describing how it bends. The second derivative \( g''(x) \) measures the rate of change of the slope, offering insights into the concavity of a graph.

• If \( g''(x) > 0 \), the graph is concave up; it looks like a cup or a U-shape, and any local minima sit in these regions.
• If \( g''(x) < 0 \), the graph is concave down; it looks like a cap or an upside-down U, which might contain local maxima.

We solve \( g''(x) = 0 \) to locate potential inflection points, places where the graph changes its concavity. For \( g(x) \), simplifying and analyzing \( g''(x) = \frac{\ln(x+1) - 2}{(x+1)^4} \) instructs us how the graph morphs its curvature between being concave up and concave down.

Critical Points

Critical points are key to understanding a function's landscape. They occur where the first derivative is zero or undefined. These points often point to local maxima, local minima, or saddle points where the function flattens out momentarily.

• By solving \( g'(x) = 0 \), we identify these critical points for \( g(x) \).
• After locating the critical points, we test the surrounding intervals to figure out the nature of each – whether each point is a local peak, dip, or flat part of the function.

This exploration of critical points enables us to draft a preliminary sketch of the graph highlighting these essential attributes, providing better clarity on how the function behaves at places of interest.

Graph Sketching

Graph sketching is a synthesis of all computational work with derivatives and concavity. It's about portraying the function \( g(x) \) visually. The key steps include:

• Identifying intervals of increase and decrease using \( g'(x) \) to see where the graph heads upwards or downwards.
• Determining concave up and concave down sections using \( g''(x) \) to reveal how the function bends over these intervals.
• Marking critical points and points of inflection, providing a skeleton of the sketch.

With all this information, you can draw a smooth, continuous curve for \( g(x) \), making sure it respects all identified intervals and key points. This visual representation aids in fully understanding the function's behavior across its domain.