Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(f^{\prime}(x)=(x-A)^{2}(x-B)^{2}, A \neq B\)

Step-by-Step Solution

Answer

The critical points of \(f'(x)\) are \(x = A\) and \(x = B\), both are potential local extrema.

Step 1: Analyze the Given Derivative Function

We have the function of the derivative, \(f'(x) = (x-A)^2(x-B)^2\). This is a polynomial function of degree 4, as each squared term \((x-A)^2\) and \((x-B)^2\) contribute a degree of 2 each, making the sum of the degrees 4.

Step 2: Determine the Critical Points

Critical points occur where the derivative is zero, \(f'(x) = 0\). Thus, set \((x-A)^2(x-B)^2 = 0\). Solving for \(x\) gives us \(x = A\) and \(x = B\). Since these factors are squared, \(A\) and \(B\) are roots of multiplicity 2.

Step 3: Evaluate the Behavior Near Critical Points

At a critical point where the derivative equals zero and has even multiplicity, the function \(f(x)\) does not change sign, indicating a local minimum or maximum (or possibly a point of inflection if higher multiplicity). Both \(x = A\) and \(x = B\) are therefore potential local extrema.

Step 4: Graphical Implications and Analysis

Given that both \(x = A\) and \(x = B\) are roots of even multiplicity, and no other terms to change the sign beyond these, the graph of \(f'(x)\) indicates the following: it will touch the x-axis, but not cross it, reinforcing the potential for local extrema at each point.

Key Concepts

Derivative

The derivative of a function is a core concept in calculus that describes how the function's value changes with respect to changes in its input. In simpler terms, it provides the slope of the function at any given point, indicating the rate of change. Calculating the derivative allows us to understand the behavior of functions, such as identifying where a function is increasing or decreasing.

Mathematically, the derivative of a function, denoted as \(f'(x)\), is calculated using rules such as the power rule, product rule, or chain rule, depending on the function's form. For instance, when given a function like \(f'(x)=(x-A)^2(x-B)^2\), understanding each component's contribution to the derivative is crucial for further analysis.

By assessing where the derivative equals zero, we can identify critical points, which are key in sketching the graph of the original function and further appreciating its overall behavior.

Polynomial function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables with constant coefficients. Each term in the polynomial function is generally of the form \(a_nx^n\), where \(a_n\) is a coefficient and \(n\) is a non-negative integer representing the degree of the term.

Understanding polynomial functions, like the given derivative \(f'(x) = (x-A)^2(x-B)^2\), is crucial because they are among the simplest yet most important types of functions in calculus. The degree of a polynomial function tells us about its complexity. Here, the function is of degree 4 because it consists of two squared terms, each contributing 2 to the total degree.

• Polynomial functions are continuous and smooth.
• They can model a variety of real-world scenarios.
• Higher-degree polynomials can have more complex curves.

Understanding their behavior through derivatives helps in finding critical points which indicate where the function reaches its extremes.

Critical points

Critical points in calculus are where the derivative of a function equals zero or is undefined. These points are significant as they can indicate where a function changes direction, marking potential maximum, minimum, or inflection points. To find critical points, we solve \(f'(x) = 0\).

In our scenario, with \(f'(x) = (x-A)^2(x-B)^2\), setting the expression equal to zero helps us find critical points: \(x = A\) and \(x = B\). These are the values where the derivative changes, and since each factor is squared, it denotes multiplicity, indicating these points are roots twice.

• Critical points with even multiplicity mean the graph touches but does not cross the x-axis.
• These points are candidates for local maxima or minima.
• Examining the behavior of the function around these points gives insight into its local behavior.

Multiplicity

Multiplicity in the context of derivatives and polynomial functions refers to the number of times a specific root occurs. When a root has a higher multiplicity, it affects the behavior of the graph of the function at that root. Specifically, if a root of a polynomial appears \(k\) times, \(k\) is its multiplicity.

In the example given, both \(x = A\) and \(x = B\) are roots of the derivative with a multiplicity of 2. This means that as the graph of the function approaches these points, it will touch the x-axis and turn back, indicating that these points are not crossings.

• Even multiplicity (like 2) means the function touches the axis and may indicate a local extremum.
• A graph showing roots with even multiplicity is often smoother at these points.
• Odd multiplicity, conversely, would mean the function crosses the x-axis.

Multiplicity provides crucial clues to the geometry of the function's graph, illustrating whether a point is a minimum, maximum, or inflection based on how the polynomial behaves around these points.