Problem 5 - Thomas Calculus

Question

Answer the following questions about the functions whose derivatives are given: \begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation} \begin{equation}f^{\prime}(x)=(x-1)(x+2)(x-3)\end{equation}

Step-by-Step Solution

Answer

Critical points: \(x = -2, 1, 3\). Increasing: \(-23\); decreasing: \(x< -2\), \(1

Step 1: Find the critical points of f

To find the critical points, we need to solve the equation \(f'(x) = 0\). Given \(f'(x) = (x-1)(x+2)(x-3)\), set it equal to zero: \[(x-1)(x+2)(x-3) = 0\]This gives the solutions \(x = 1\), \(x = -2\), and \(x = 3\). These are the critical points of \(f\).

Step 2: Determine intervals of increase or decrease

The sign of \(f'(x)\) determines whether \(f\) is increasing or decreasing. Evaluate \(f'(x)\) in the intervals defined by the critical points: - For \(-\infty < x < -2\), choose \(x = -3\): \((x-1)(x+2)(x-3) > 0\), so \(f\) is decreasing.- For \(-2 < x < 1\), choose \(x = 0\): \((x-1)(x+2)(x-3) < 0\), so \(f\) is increasing.- For \(1 < x < 3\), choose \(x = 2\): \((x-1)(x+2)(x-3) > 0\), so \(f\) is decreasing.- For \(x > 3\), choose \(x = 4\): \((x-1)(x+2)(x-3) > 0\), so \(f\) is increasing.

Step 3: Identify local maxima and minima

To determine if critical points are local maxima or minima, observe sign changes in \(f'(x)\):- At \(x = -2\), \(f'(x)\) changes from negative to positive, indicating a local minimum.- At \(x = 1\), \(f'(x)\) changes from positive to negative, indicating a local maximum.- At \(x = 3\), \(f'(x)\) changes from negative to positive, indicating a local minimum.

Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus. They help us understand how a function changes at any given point. For a function \( f(x) \), its derivative, denoted as \( f'(x) \), tells us the rate at which \( f(x) \) is changing with respect to \( x \).

To find critical points, we need to set the derivative equal to zero and solve for \( x \). These are the points where the slope of the function is zero, indicating that the function may have a peak or a valley. In the given problem, the derivative \( f'(x) = (x-1)(x+2)(x-3) \) was used to find the critical points by solving \( (x-1)(x+2)(x-3) = 0 \), which gives \( x = 1, -2, \) and \( 3 \).

Thus, the derivative helps us identify important features of the function, such as slopes, critical points, and potential peaks and troughs.

Intervals of Increase and Decrease

Once we have the critical points, we can determine the intervals where the function is increasing or decreasing. This involves examining the sign of the derivative in each interval determined by these critical points.

• For \( -\infty < x < -2 \), the derivative is positive, indicating that the function \( f(x) \) is decreasing.
• For \( -2 < x < 1 \), the derivative is negative, indicating that the function is increasing.
• For \( 1 < x < 3 \), the derivative again becomes positive, showing that \( f(x) \) is decreasing.
• For \( x > 3 \), the derivative is positive, and the function is increasing.

By understanding where a function increases or decreases, we gain insights into its overall behavior and shape, which helps in predicting how it might look when graphed.

Local Maxima and Minima

Local maxima and minima are significant points on a graph where a function reaches a high or low value in a specific neighborhood.

At a local maximum, the function changes from increasing to decreasing, implying a peak. Similarly, at a local minimum, the function changes from decreasing to increasing, signifying a valley.

• At \( x = -2 \), \( f'(x) \) changes from negative to positive, indicating a local minimum.
• At \( x = 1 \), \( f'(x) \) changes from positive to negative, indicating a local maximum.
• At \( x = 3 \), \( f'(x) \) changes from negative to positive, indicating another local minimum.

Recognizing these points is essential for understanding the function's graph, especially in terms of optimizing and analyzing natural systems or processes.

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Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus. They help us understand how a function changes at any given point. For a function \( f(x) \), its derivative, denoted as \( f'(x) \), tells us the rate at which \( f(x) \) is changing with respect to \( x \).

To find critical points, we need to set the derivative equal to zero and solve for \( x \). These are the points where the slope of the function is zero, indicating that the function may have a peak or a valley. In the given problem, the derivative \( f'(x) = (x-1)(x+2)(x-3) \) was used to find the critical points by solving \( (x-1)(x+2)(x-3) = 0 \), which gives \( x = 1, -2, \) and \( 3 \).

Thus, the derivative helps us identify important features of the function, such as slopes, critical points, and potential peaks and troughs.

Intervals of Increase and Decrease

Once we have the critical points, we can determine the intervals where the function is increasing or decreasing. This involves examining the sign of the derivative in each interval determined by these critical points.

• For \( -\infty < x < -2 \), the derivative is positive, indicating that the function \( f(x) \) is decreasing.
• For \( -2 < x < 1 \), the derivative is negative, indicating that the function is increasing.
• For \( 1 < x < 3 \), the derivative again becomes positive, showing that \( f(x) \) is decreasing.
• For \( x > 3 \), the derivative is positive, and the function is increasing.

By understanding where a function increases or decreases, we gain insights into its overall behavior and shape, which helps in predicting how it might look when graphed.

Local Maxima and Minima

Local maxima and minima are significant points on a graph where a function reaches a high or low value in a specific neighborhood.

At a local maximum, the function changes from increasing to decreasing, implying a peak. Similarly, at a local minimum, the function changes from decreasing to increasing, signifying a valley.

• At \( x = -2 \), \( f'(x) \) changes from negative to positive, indicating a local minimum.
• At \( x = 1 \), \( f'(x) \) changes from positive to negative, indicating a local maximum.
• At \( x = 3 \), \( f'(x) \) changes from negative to positive, indicating another local minimum.

Recognizing these points is essential for understanding the function's graph, especially in terms of optimizing and analyzing natural systems or processes.