Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.3.1b

Chapter 4, Problem 4.3.1b

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

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.

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.

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.

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.

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Derivatives Applied To Acceleration Example 2

Related Practice

Textbook Question

114. Parabolas

b. When is the parabola concave up? Concave down?

Textbook Question

Identifying Extrema

In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).

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b. Either use the graph to determine which intervals f is positive on and which intervals f is negative on, or explain why this information cannot be determined from the graph.

Textbook Question

Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local maximum at (1, 1) and a local minimum at (3, 3).

Textbook Question

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = (x − 2)²ᐟ³.

b. Show that the only local extreme value of f occurs at x = 2.

Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

-(1/2)x⁻³ᐟ²