Problem 19 - Essential Calculus Early Transcendentals

Question

Sketch the graph of a function that satisfies all of the given conditions. \(f^{\prime}(0)=f^{\prime}(2)=f^{\prime}(4)=0,\) \(f^{\prime}(x)>0\) if \(x < 0\) or \(2 < x < 4,\) \(f^{\prime}(x) < 0\) if \(0 < x < 2\) or \(x>4,\) \(f^{\prime \prime}(x)>0\) if \(1 < x <3, \quad f^{\prime \prime}(x) <0\) if \(x<1\) or \(x>3\)

Step-by-Step Solution

Answer

Graph has local max at \( x=0 \); local mins at \( x=2, 4 \); sketch accordingly.

Step 1: Identify Critical Points

Identify the points where the derivative is zero, which are given as: \( x = 0 \), \( x = 2 \), and \( x = 4 \). These points are the critical points where the function may have local maxima or minima.

Step 2: Determine Intervals of Increasing and Decreasing

Analyze the intervals where the first derivative \( f'(x) \) is greater than or less than zero. The function is increasing on the intervals \( (-\infty, 0) \) and \( (2, 4) \), and decreasing on the intervals \( (0, 2) \) and \( (4, \infty) \).

Step 3: Identify Concave Up and Down Intervals

Use the second derivative \( f''(x) \) to determine concavity: the function is concave down (\( f''(x) < 0 \)) on \( (-\infty, 1) \) and \( (3, \infty) \), and concave up (\( f''(x) > 0 \)) on \( (1, 3) \).

Step 4: Sketch the Graph

Using the information from Steps 1-3, sketch a curve that passes through points where the first derivative is zero (local maxima at \( x = 0 \) and local minima at \( x = 2 \) and \( x = 4 \)). The function should be increasing before \( x = 0 \) and between \( x = 2 \) and \( x = 4 \), and decreasing between \( x = 0 \) and \( x = 2 \) and after \( x = 4 \). The graph should also be concave down until \( x = 1 \), concave up between \( x = 1 \) and \( x = 3 \), and concave down again after \( x = 3 \).

Key Concepts

Critical Points

Critical points are special values of a function where its first derivative is zero. These points potentially indicate a change in the function's behavior—either a local maximum or minimum. For the function given in the exercise, the critical points are at

• \( x = 0 \)
• \( x = 2 \)
• \( x = 4 \)

At these points, the function's slope is zero, meaning it's neither increasing nor decreasing momentarily. To determine whether these points are local minima or maxima, further analysis, such as using the second derivative test or a graph sketch, can be employed. In this exercise, identifying these critical points is the first step towards understanding how the graph behaves.

Increasing and Decreasing Intervals

To determine where a function is increasing or decreasing, we need to look at the sign of its first derivative. Whenever the first derivative \( f'(x) \) is greater than zero, the original function is increasing. Conversely, when \( f'(x) \) is less than zero, the function is decreasing. According to the problem, the function behaves in several distinct intervals:

• It is increasing in the intervals \( (-\infty, 0) \) and \( (2, 4) \)
• It is decreasing in the intervals \( (0, 2) \) and \( (4, \infty) \)

These intervals are crucial for sketching the function correctly, as they highlight where the function goes up and down.

Concavity

The concavity of a function tells us how the curve bends. This characteristic is determined using the second derivative. If \( f''(x) > 0 \), the function is concave up, forming a U-shape. On the other hand, if \( f''(x) < 0 \), it is concave down, forming an upside-down U. For our function:

• The graph is concave down on the intervals \( (-\infty, 1) \) and \( (3, \infty) \)
• It is concave up on the interval \( (1, 3) \)

By understanding these intervals, you can better sketch the curve, ensuring it properly bends upwards or downwards between critical points.

Derivative Analysis

Derivative analysis encompasses both the first and second derivatives to provide a comprehensive understanding of the function's behavior. The first derivative, \( f'(x) \), helps us understand where the function is increasing or decreasing. The zero points of \( f'(x) \) highlight the critical points and potential turning places for maxima or minima. Meanwhile, the second derivative, \( f''(x) \), gives insights into the concavity, telling us how the graph curves in the given sections.Graph sketching leverages these analyses to visually represent the function:

• Start by plotting critical points as turning spots where the function momentarily flattens
• Examine intervals of increasing and decreasing trends for upward or downward slopes
• Apply concavity determined from \( f''(x) \) to guide the bending of the graph correctly

Together, these steps provide a holistic understanding enabling accurate graph sketching, revealing the nature of the function in question.