Problem 33 - Essential Calculus

Question

Find the critical numbers of \(f\) (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. $$ f(x)=x+\frac{1}{x} $$

Step-by-Step Solution

Answer

The critical numbers of the function are \(\pm 1\). The function is decreasing on the intervals \((-∞, -1)\) and \((1, ∞)\), and increasing on the interval \((-1, 1)\). The function has a relative maximum at x = -1 and a relative minimum at x = 1.

Step 1: Compute the derivative of the function

To compute the derivative of the function \(f(x) = x + \frac{1}{x}\), we use the power rule \(d/dx [x^n] = nx^{n-1}\) and the rule \(d/dx [1/x] = -1/x^2\). This gives \(f'(x) = 1 - \frac{1}{x^2}\).

Step 2: Find the critical numbers of the function

Critical numbers of a function are those x-values where the derivative equals zero, or the derivative is undefined. To find the critical numbers of this function, set \(f'(x) = 0\), which gives \(1 - \frac{1}{x^2} = 0\). Solving for \(x\), we get \(x = \pm 1\). These are the critical numbers for the function.

Step 3: Perform the first derivative test

By choosing test points to the left and the right of each critical number and evaluating the derivative at those points, we can determine whether the function increases or decreases. If the derivative is positive, the function increases (upwards). If the derivative is negative, the function decreases (downwards). For \(x < -1, x = -1, -1 < x < 1, x = 1, x > 1\), we choose test points (-2, -1, 0, 1, 2), and get \(f'(-2) = -3, f'(-1) = 0, f'(0) = 1, f'(1) = 0, f'(2) = -3\). Thus, the function is decreasing on the interval (-∞, -1), increasing on (-1, 1), and decreasing on (1, ∞). Hence, there is a relative maximum at x = -1 and a relative minimum at x = 1.

Step 4: Graph the function

Use a graphing utility to confirm the results. The function shows a relative maxima at x = -1 and a relative minima at x = 1, which matches our calculations.

Key Concepts

First Derivative Test

The First Derivative Test is a method used to determine whether a critical number corresponds to a relative maximum, minimum, or neither. It involves analyzing the sign changes in the derivative of a function. If the derivative changes from positive to negative at a critical point, the function has a relative maximum there. If the derivative changes from negative to positive, the function has a relative minimum. Here's how you can use the First Derivative Test:

• Identify the critical numbers by solving for when the derivative equals zero or is undefined.
• Choose test points in the intervals between and around the critical numbers.
• Evaluate the derivative at these test points to determine the sign (positive or negative).
• Conclude whether each critical number is a point of relative maximum, minimum, or neither based on the sign changes.

This test provides insight into the behavior of functions, helping us understand where they rise and fall.

Increasing and Decreasing Functions

Functions can be described as increasing or decreasing over specific intervals. An increasing function means that as you move from left to right on a graph, the values of the function get higher. Conversely, a decreasing function's values drop as you move from left to right.To determine the intervals where a function is increasing or decreasing, examine its derivative:

• If the derivative is positive over an interval, the function is increasing there.
• If the derivative is negative over an interval, the function is decreasing there.

In our specific case with the function \(f(x) = x + \frac{1}{x}\), we calculated that it's increasing on the interval \((-1, 1)\) and decreasing on \((-\infty, -1)\) and \((1, \infty)\). This was concluded by evaluating the first derivative at selected test points.

Relative Extrema

Relative extrema refer to the points on the graph of a function where the function reaches a relative maximum or minimum. These points are crucial in understanding the shape and features of the graph.To find relative extrema:

• Identify the critical numbers where the derivative equals zero or is undefined.
• Use the First Derivative Test to determine the nature of these points.

For the function \(f(x) = x + \frac{1}{x}\), we found that there is a relative maximum at \(x = -1\) and a relative minimum at \(x = 1\). This was verified by observing how the signs of the first derivative change around these critical numbers.

Derivative Calculation

Calculating the derivative of a function is essential for analyzing its properties, such as finding critical numbers, determining intervals of increase or decrease, and locating relative extrema.Here's a simple guide to calculating derivatives:

• Apply derivative rules: for a power function \(x^n\), the derivative is \(nx^{n-1}\).
• For a function like \(\frac{1}{x}\), use the rule \(d/dx [1/x] = -1/x^2\).

In our exercise, the derivative of \(f(x) = x + \frac{1}{x}\) is calculated to be \(f'(x) = 1 - \frac{1}{x^2}\). This derivative forms the basis for finding critical points, determining where the function increasing or decreasing, and identifying relative extrema.