Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

Critical points: x = 1, 2, 3, 4; increasing on (-∞, 3) U (3, ∞), decreasing on (2, 3).

Step 1: Identify the Given Function

We are given the derivative of a function, which is \( f'(x) = (x-1)^2(x-2)^2(x-3)(x-4) \). This expression implies that it is a polynomial function.

Step 2: Determine Critical Points

Critical points occur where the derivative is zero. Solve \( f'(x) = 0 \): \( (x-1)^2(x-2)^2(x-3)(x-4) = 0 \). The critical points are \( x = 1, 2, 3, 4 \).

Step 3: Analyze Multiplicity

The critical points \( x = 1 \) and \( x = 2 \) have a multiplicity of 2, meaning the graph will touch but not cross the x-axis at these points. The points \( x = 3 \) and \( x = 4 \) have a multiplicity of 1, meaning the graph will cross the x-axis at these points.

Step 4: Determine Behavior Around Critical Points

For multiplicity of 2 at \( x = 1, 2 \), the function shows a local minimum or maximum (touch points). At \( x = 3, 4 \) with multiplicity 1, the function changes from increasing to decreasing or vice versa (cross points).

Step 5: Determine Increasing/Decreasing Intervals

To find where \( f(x) \) increases or decreases, test intervals between critical points. Choose test points, such as \( x = 0, 1.5, 2.5, 3.5, 5 \), and check the sign of \( f'(x) \).- For \( x < 1 \), test \( x = 0 \): \( f'(0) = 1 \times 4 \times 3 \times 4 > 0 \) (increasing).- For \( 1 < x < 2 \), test \( x = 1.5 \): the product is positive (increasing).- For \( 2 < x < 3 \), test \( x = 2.5 \): the product is negative (decreasing).- For \( 3 < x < 4 \), test \( x = 3.5 \): the product is positive (increasing).- For \( x > 4 \), test \( x = 5 \): the product is positive (increasing).

Key Concepts

Polynomial Functions

Polynomial functions are one of the most foundational concepts in calculus. They are mathematical expressions involving a sum of powers in one or more variables, with constant coefficients. For example, a polynomial function can be as simple as \( x^2 + 2x + 1 \). These functions appear frequently throughout mathematics because of their nice properties and simplicity.

In higher mathematics, especially calculus, we frequently encounter polynomial functions because they are easy to differentiate and integrate. The derivative of a polynomial function results in another polynomial function. This ease of use makes them excellent tools for modeling and solving problems.

When analyzing the function \( f'(x) = (x-1)^2(x-2)^2(x-3)(x-4) \), you can see it is expressed in a factored form. The significance of finding the derivative of this polynomial is that it helps us to find critical points, which tell us a lot about the function's behavior.

Derivative Analysis

Derivative analysis is a powerful tool to understand how a function behaves. When you take the derivative of a function, you are essentially finding the slope of the tangent line to the function at any given point. This gives insight into the function's rate of change.

For the polynomial derivative \( f'(x) = (x-1)^2(x-2)^2(x-3)(x-4) \), analyzing the derivative will help determine where the original function is increasing, decreasing, or changing direction. By setting \( f'(x) = 0 \), you find the critical points. These points indicate where the slope is zero, typically corresponding to local maxima, minima, or inflection points.

• Local Maximum: Point where the function changes from increasing to decreasing.
• Local Minimum: Point where the function changes from decreasing to increasing.
• Inflection Point: Point where the function changes its concavity.

Derivatives not only provide information about the slope, but their analysis enables you to predict and graph the original function's behavior.

Multiplicity of Roots

The concept of multiplicity of roots relates to how many times a particular root occurs in a function. When you have an expression such as \((x-a)^n\), the value \(a\) is a root with a multiplicity \(n\). In the polynomial \( (x-1)^2(x-2)^2(x-3)(x-4) \), you can see different behavior at different critical points based on their multiplicities.

• Multiplicity 1: This indicates a simple root. The graph will cross the x-axis at this point, suggesting a change from increasing to decreasing or vice versa.
• Multiplicity 2: The root is repeated. The graph touches the x-axis at these points, indicating that the function neither increases nor decreases but momentarily levels off, typically forming a local maximum or minimum.

Understanding the multiplicity helps in sketching the graph, as it provides clues about how the function behaves near its roots.

Increasing and Decreasing Intervals

Identifying the intervals where a function is increasing or decreasing is crucial for understanding its behavior. This process involves testing the sign of the derivative in different intervals made by the critical points. The sign of the derivative tells us whether the function is increasing or decreasing.

• Increasing Interval: If \( f'(x) > 0 \), the function is increasing in that interval.
• Decreasing Interval: If \( f'(x) < 0 \), the function is decreasing in that interval.

In the example with the polynomial derivative \( f'(x) = (x-1)^2(x-2)^2(x-3)(x-4) \), by examining specific points within each interval — such as 0, 1.5, 2.5, 3.5, and 5 — you can reveal intervals of increase (\( x < 1, 1 < x < 2, 3 < x < 4, x > 4 \)) and decrease (\( 2 < x < 3 \)).

Understanding these intervals helps in creating an accurate graph and provides a clearer picture of the function's overall behavior.