Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.3.38
Chapter 4, Problem 4.3.38

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = x ln x - 2x + 3 on (0,∞)

Verified step by step guidance
1
First, find the derivative of the function f(x) = x ln x - 2x + 3. Use the product rule for the term x ln x, which states that if u(x) = x and v(x) = ln x, then the derivative is u'(x)v(x) + u(x)v'(x).
Calculate the derivative: f'(x) = (1)(ln x) + (x)(1/x) - 2. Simplify this to get f'(x) = ln x + 1 - 2.
Simplify the expression for the derivative: f'(x) = ln x - 1.
Determine the critical points by setting the derivative equal to zero: ln x - 1 = 0. Solve for x to find the critical points.
Analyze the sign of f'(x) on the intervals determined by the critical points. If f'(x) > 0, the function is increasing on that interval; if f'(x) < 0, the function is decreasing. Use test points in each interval to determine the sign of f'(x).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

Key Concepts

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

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental tool in calculus for determining the slope of the tangent line to the curve at any point. For a function to be increasing, its derivative must be positive, while a negative derivative indicates that the function is decreasing.
Recommended video:
05:44
Derivatives

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are essential for analyzing the behavior of the function, as they can indicate potential local maxima, minima, or points of inflection. To determine intervals of increase or decrease, one must find these critical points and test the sign of the derivative in the intervals they create.
Recommended video:
04:50
Critical Points

First Derivative Test

The First Derivative Test is a method used to determine whether a function is increasing or decreasing on specific intervals. By evaluating the sign of the derivative before and after each critical point, one can conclude whether the function transitions from increasing to decreasing or vice versa. This test provides a systematic approach to understanding the function's overall behavior.
Recommended video:
07:09
The First Derivative Test: Finding Local Extrema
Related Practice
Textbook Question

49–54. {Use of Tech} Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.


ƒ(x) = 3x⁴ + 4x³ - 12x²

Textbook Question

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = ln x - x² + 3x - 1

Textbook Question

Approximating changes


Approximate the change in the volume of a right circular cylinder of fixed radius r = 20 cm when its height decreases from h = 12 to h = 11.9 cm (V(h) = πr²h).

1
views
Textbook Question

{Use of Tech} Newton’s method and curve sketching Use Newton’s method to find approximate answers to the following questions.

Where are the inflection points of f(x) = (9/5)x⁵ - (15/2)x⁴ + (7/3)x³ + 30x² + 1 located?

Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_u→ π/4 (tan u - cot u) / (u - π/4)

Textbook Question

Graphing functions Use the guidelines of this section to make a complete graph of f.


f(x) = x² - 6x