Problem 20 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

In Exercises 17 through 20, sketch the graph of a function \(f\) that has all the given properties. a. \(f^{\prime}(x)>0\) when \(x<1\) b. \(f^{\prime}(x)<0\) when \(x>1\) c. \(f^{\prime \prime}(x)>0\) when \(x<1\) and when \(x>1\) d. \(f^{\prime}(1)\) is undefined.

Step-by-Step Solution

Answer

Increase and concave up before x = 1, sharp peak at x = 1, decrease and concave up after x = 1.

Step 1: Analyze the derivative sign changes

Identify where the first derivative, \( f' (x) \) changes sign. According to the given information, \( f' (x) > 0 \) for \( x < 1 \) and \( f' (x) < 0 \) for \( x > 1 \). This suggests that the function is increasing when \( x < 1 \) and decreasing when \( x > 1 \). Therefore, it has a local maximum at \( x = 1 \).

Step 2: Analyze the second derivative

Examine where the second derivative, \( f'' (x) \), is positive. The given information tells us that \( f'' (x) > 0 \) for \( x < 1 \) and \( x > 1 \). A positive second derivative indicates that the function is concave up on both sides of \( x = 1 \).

Step 3: Consider the point where derivative is undefined

Note that \( f'(1) \) is undefined. This means there is a sharp point or cusp at \( x = 1 \).

Step 4: Sketch the graph of the function

Using the information from Steps 1-3, sketch the graph of the function. The function should be increasing and concave up before \( x = 1 \), reach a peak at \( x = 1 \), and then decrease while being concave up after \( x = 1 \). Ensure the graph has a sharp point at \( x = 1 \) to represent the undefined derivative.

Key Concepts

First Derivative

The first derivative of a function, denoted as \( f'(x) \), tells us about the function's rate of change. When \( f'(x) > 0 \), the function is increasing, meaning the graph goes upward as you move from left to right. When \( f'(x) < 0 \), the function is decreasing, and the graph goes downward. In our example, the derivative is positive when \( x < 1 \) and negative when \( x > 1 \). This change signifies a local maximum at \( x = 1 \), where the function switches from increasing to decreasing.

Second Derivative

The second derivative, \( f''(x) \), provides information about the concavity of the function. If \( f''(x) > 0 \), the graph is concave up, resembling a U-shape; if \( f''(x) < 0 \), the graph is concave down, resembling an upside-down U. In this case, \( f''(x) > 0 \) for both \( x < 1 \) and \( x > 1 \). Therefore, the function is concave up on both sides of \( x = 1 \).

Local Maximum

A local maximum occurs at a point where the function changes from increasing to decreasing. In other words, it is a peak in the graph. For our specific function, this occurs at \( x = 1 \). The function increases up to this point and then starts to decrease, making \( x = 1 \) a peak. Remember, to determine this, you look for the point where the first derivative changes sign from positive to negative.

Concavity

Concavity tells us how the graph bends. If a function is concave up (\( f''(x) > 0 \)), it bends upwards, creating a valley-like shape. If it is concave down (\( f''(x) < 0 \)), it bends downwards, creating a hill-like shape. Our function is concave up on both sides of \( x = 1 \), indicated by the positive second derivative in those intervals. Understanding concavity helps in sketching and visualizing the function's overall shape accurately.

Undefined Derivative

An undefined derivative at a point means the slope of the tangent line to the graph is not well-defined at that point. This can occur at a sharp point or cusp. For our function, \( f'(1) \) is undefined, suggesting a cusp at \( x = 1 \). In graphing terms, this means that the function has a sharp turn or point at \( x = 1 \), which is an important characteristic to accurately represent when sketching the graph.