Analyzing Functions from Derivatives


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


b. On what open intervals is f increasing or decreasing?


f′(x) = x(x − 1)

Verified step by step guidance

1. To determine where the function f is increasing or decreasing, we need to analyze the sign of its derivative f'(x). The function is increasing where f'(x) > 0 and decreasing where f'(x) < 0.
2. First, find the critical points by setting the derivative equal to zero: f'(x) = x(x - 1) = 0. Solve for x to find the critical points.
3. The solutions to the equation x(x - 1) = 0 are x = 0 and x = 1. These are the critical points where the derivative changes sign.
4. Next, determine the sign of f'(x) on the intervals defined by the critical points: (-∞, 0), (0, 1), and (1, ∞). Choose a test point from each interval and substitute it into f'(x) to determine the sign.
5. Based on the sign of f'(x) in each interval, conclude where the function f is increasing or decreasing. If f'(x) > 0 in an interval, f is increasing there; if f'(x) < 0, f is decreasing.

Key Concepts

Derivative and Critical Points

The derivative of a function, f'(x), provides information about the slope of the tangent line at any point x. Critical points occur where f'(x) = 0 or is undefined, indicating potential maxima, minima, or points of inflection. Identifying these points is crucial for determining intervals of increase or decrease in the function.

Increasing and Decreasing Intervals

A function is increasing on an interval if its derivative is positive over that interval, meaning the function's slope is upward. Conversely, it is decreasing if the derivative is negative, indicating a downward slope. Analyzing the sign of f'(x) across different intervals helps identify where the function is increasing or decreasing.

Sign Analysis of Derivatives

Sign analysis involves determining the sign of the derivative f'(x) over various intervals. For f'(x) = x(x - 1), factorization reveals critical points at x = 0 and x = 1. By testing values in intervals around these points, one can ascertain where f'(x) is positive or negative, thus identifying the function's behavior in terms of increasing or decreasing.