Problem 1 - University Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Critical points at \(x = 0\) and \(x = 1\). \(f\) is increasing on \((-\infty, 0)\) and \((1, \infty)\), decreasing on \((0, 1)\). Local max at \(x = 0\), min at \(x = 1\).

Step 1: Find the Critical Points

Critical points occur where the derivative is zero or undefined. For the function with derivative \(f'(x)=x(x-1)\), set it equal to zero: \(x(x-1) = 0\). This gives the critical points \(x = 0\) and \(x = 1\).

Step 2: Determine Intervals for Increasing/Decreasing

To find where the function is increasing or decreasing, analyze the sign of \(f'(x) = x(x-1)\) on the intervals determined by the critical points: \((-\infty, 0)\), \((0, 1)\), and \((1, \infty)\).- On \((-\infty, 0)\): Choose \(x = -1\). \(f'(-1) = (-1)((-1)-1) = 2\) (positive), so \(f\) is increasing.- On \((0, 1)\): Choose \(x = 0.5\). \(f'(0.5) = 0.5 (0.5 -1) = -0.25\) (negative), so \(f\) is decreasing.- On \((1, \infty)\): Choose \(x = 2\). \(f'(2) = 2(2-1) = 2\) (positive), so \(f\) is increasing.

Step 3: Identify Local Extrema

Local extrema occur at critical points where the derivative changes sign. - At \(x = 0\): \(f\) changes from increasing to decreasing, so there is a local maximum at \(x = 0\).- At \(x = 1\): \(f\) changes from decreasing to increasing, so there is a local minimum at \(x = 1\).

Key Concepts

Critical Points

Critical points are a fundamental concept in calculus that help us understand the behavior of a function. A critical point is a value of the variable, say \(x\), at which the derivative of the function is either zero or undefined. These points are crucial because they are potential spots where the function could achieve a local maximum or minimum. When examining the function's graph, critical points often indicate where changes in the slope occur. To find the critical points of a function with a given derivative, like \(f'(x) = x(x-1)\), you need to solve for \(x\) when \(f'(x) = 0\). Here, solving the equation \(x(x-1) = 0\) results in critical points at \(x = 0\) and \(x = 1\). Remember:

• If the derivative is zero, the function may be flat (neither increasing nor decreasing) at those points.
• Where the derivative is undefined, the function might have a cusp or vertical tangency.
• Critical points do not guarantee local maxima or minima but are key candidates.

Increasing and Decreasing Functions

Understanding whether a function is increasing or decreasing helps to analyze its behavior and shape. The intervals where a function is increasing or decreasing can be found by examining the sign of its derivative.For the function with derivative \(f'(x) = x(x-1)\):

• If the derivative \(f'(x) > 0\), the function \(f\) is increasing on that interval.
• If the derivative \(f'(x) < 0\), the function \(f\) is decreasing on that interval.

To apply this, test points within the intervals determined by critical points — in this case,

1. \((-\infty, 0)\): Choosing \(x = -1\), we find \(f'(-1) = 2\), which is positive, indicating that \(f\) is increasing.
2. \((0, 1)\): Choosing \(x = 0.5\), we find \(f'(0.5) = -0.25\), which is negative, indicating that \(f\) is decreasing.
3. \((1, \infty)\): Choosing \(x = 2\), we find \(f'(2) = 2\), which is positive, indicating that \(f\) is increasing again.

These tests provide a clearer picture of how the function behaves between and around critical points.

Derivative

The derivative is a powerful tool in calculus representing the rate at which a function is changing at any given point. In simpler terms, it tells us the slope of the function at a particular point. The derivative is fundamental when analyzing and studying the dynamics of functions. For a function \(f(x)\), its derivative is written as \(f'(x)\) or \(\frac{df}{dx}\). Understanding the derivative helps to:

• Identify the direction in which the function is heading (increasing or decreasing).
• Locate critical points which are potential spots of local maxima or minima.
• Establish the concavity and overall shape of a function's graph.

In our exercise, the derivative given is \(f'(x) = x(x-1)\). Solving for when \(f'(x) = 0\) helps find critical points. The behavior of \(f'(x)\) over intervals determined by these points explains the increasing or decreasing nature of \(f(x)\), showcasing how invaluable derivatives are in understanding and interpreting the characteristics of functions.