Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:


a. What are the critical points of f?


f′(x) = 1− 4/x², x ≠ 0

Verified step by step guidance

1. To find the critical points of the function f, we need to determine where the derivative f'(x) is equal to zero or undefined. Critical points occur where the derivative changes sign or is undefined.
2. Given the derivative f'(x) = 1 - 4/x², we first set f'(x) equal to zero to find where the derivative changes sign: 1 - 4/x² = 0.
3. Solve the equation 1 - 4/x² = 0 for x. This involves isolating x² by adding 4/x² to both sides, resulting in 1 = 4/x².
4. Next, solve for x² by multiplying both sides by x², giving x² = 4. Then, take the square root of both sides to find the values of x: x = ±2.
5. Since x ≠ 0, the critical points are x = 2 and x = -2. These are the points where the derivative is zero, indicating potential maxima, minima, or points of inflection.

Key Concepts

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are significant because they can indicate potential local maxima, minima, or points of inflection. To find critical points, set the derivative equal to zero and solve for x, or identify where the derivative does not exist.

Derivative Analysis

Analyzing the derivative of a function helps determine the behavior of the original function. The sign of the derivative indicates whether the function is increasing or decreasing. In this context, f′(x) = 1 - 4/x² must be analyzed to find where it equals zero or is undefined, revealing critical points.

Rational Functions

A rational function is a ratio of two polynomials. The derivative given, f′(x) = 1 - 4/x², is a rational function. Understanding how to manipulate and solve rational functions is crucial for finding where the derivative equals zero or is undefined, which helps identify critical points.