Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(f^{\prime}(x)=(x-1)^{2}(x-2)^{2}(x-3)^{2}(x-4)^{2}\)

Step-by-Step Solution

Answer

The derivative is zero at \( x = 1, 2, 3, 4 \) with flat tangents and no local extrema due to even multiplicity.

Step 1: Understand the Problem

The problem requires us to analyze the function's derivative, particularly its behavior and critical points, given by \( f'(x) = (x-1)^2(x-2)^2(x-3)^2(x-4)^2 \).

Step 2: Identify Critical Points

A critical point occurs where the derivative is zero or undefined. Here, the derivative \( f'(x) \) is zero when any term in the product is zero. Thus, \( f'(x) = 0 \) at \( x = 1, 2, 3, \) and \( 4 \).

Step 3: Check Multiplicity of Zeros

Each factor in \( (x-1)^2(x-2)^2(x-3)^2(x-4)^2 \) is squared, indicating that each critical point has a multiplicity greater than one. High multiplicity suggests the function does not change sign at each critical point.

Step 4: Determine the Nature of Critical Points Using First Derivative Test

Since each critical point has an even multiplicity, the sign of \( f'(x) \) does not change around these points. Hence, these are points of inflection, but not maxima or minima.

Step 5: Conclude on the Characteristics of the Function

Given the nature of the derivative and the multiplicity of zeros, the function has a flat tangent (i.e., horizontal tangent line) at the critical points, exhibiting no local minima or maxima at \( x = 1, 2, 3, \) and \( 4 \).

Key Concepts

Critical Points

Critical points are locations on a graph of a function where the derivative is zero or undefined. These points are significant because they can help identify potential maximums, minimums, or points of inflection of the function. In many problems, focusing on critical points helps to understand the overall behavior of the function. For the function given by its derivative \( f'(x) = (x-1)^2(x-2)^2(x-3)^2(x-4)^2 \), the critical points occur where the derivative equals zero. This happens when any factor of the product equals zero:

• \( x = 1 \)
• \( x = 2 \)
• \( x = 3 \)
• \( x = 4 \)

These values are the roots of the derivative, signifying critical points of the original function. Understanding the behavior of the function around these points is crucial for analyzing the function's graph.

First Derivative Test

The first derivative test is a valuable method in calculus for determining the nature of critical points. It helps decide whether a critical point is a local maximum, a local minimum, or neither. Simply put, this test involves observing how the derivative's sign changes on either side of a critical point. By evaluating such changes, we can conclude the nature of the critical point:

• If the derivative changes from positive to negative, the critical point is a local maximum.
• If it changes from negative to positive, it indicates a local minimum.
• If there is no change in sign, the point is a point of inflection.

In the example function, each critical point has an even multiplicity, implying that the derivative does not change sign around these points. Therefore, none of these critical points are maxima or minima. They are points of inflection, where the function's graph has a flat tangent line.

Zeros Multiplicity

Multiplicity of zeros is an important concept when analyzing polynomials and their derivatives. It refers to the number of times a particular root is repeated in a polynomial equation. The multiplicity affects the behavior of the function at that root. For instance, if a root has a multiplicity of 2 or more (an even number), it suggests that the function's graph touches or is tangent to the x-axis at this point, rather than crossing it. This results in a less noticeable change in direction.In the derivative \( f'(x) = (x-1)^2(x-2)^2(x-3)^2(x-4)^2 \), each factor is squared, giving the roots \( x = 1, 2, 3, \) and \( 4 \) a multiplicity of 2. This even multiplicity means that, at each of these critical points, the derivative does not change sign, and the function remains flat. Thus, these critical points do not indicate any local maxima or minima, but rather points of inflection without the graph of the function changing direction significantly.