Textbook Question
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
iii. y = x³ − 3x² + 4 = (x + 1)(x − 2)²
1
views
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.3.51a
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
g(x) = (x − 2) / (x²−1), 0 ≤ x < 1
Verified step by step guidance1
First, find the derivative of the function g(x) = \( \frac{x - 2}{x^2 - 1} \). Use the quotient rule, which states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{u'v - uv'}{v^2} \). Here, \( u = x - 2 \) and \( v = x^2 - 1 \).
Calculate the derivatives: \( u' = 1 \) and \( v' = 2x \). Substitute these into the quotient rule formula to find \( g'(x) \).
Set the derivative \( g'(x) \) equal to zero to find critical points. Solve the equation \( \frac{(x^2 - 1) - (x - 2)(2x)}{(x^2 - 1)^2} = 0 \) to find the values of x where the derivative is zero.
Check the critical points and endpoints within the domain \( 0 \leq x < 1 \) to determine if they are local extrema. Evaluate the function g(x) at these points.
Use the first or second derivative test to classify the critical points as local minima, maxima, or neither. The first derivative test involves checking the sign of \( g'(x) \) around the critical points, while the second derivative test involves evaluating \( g''(x) \) at the critical points.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Local Extrema
Local extrema refer to the points in a function where it reaches a local maximum or minimum within a specific interval. To identify these points, one typically examines the function's derivative to find critical points where the derivative is zero or undefined, and then uses the second derivative test or other methods to determine the nature of these points.
Recommended video:
05:58
Finding Extrema Graphically
Critical Points
Critical points are values of x where the derivative of a function is either zero or undefined. These points are potential candidates for local extrema. To find them, calculate the derivative of the function and solve for x where the derivative equals zero or does not exist. Analyzing these points helps in determining the behavior of the function around them.
Recommended video:
04:50Critical Points
Domain Restrictions
Domain restrictions define the set of x-values for which a function is considered. In this problem, the domain is 0 ≤ x < 1, meaning the function is only analyzed within this interval. Understanding domain restrictions is crucial as they limit where extrema can occur and affect the behavior of the function, especially near boundaries or points of discontinuity.
Recommended video:
5:10Finding the Domain and Range of a Graph
Related Practice
Textbook Question
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
iv. y = x³ − 33x² + 216x = x(x - 9)(x − 24)
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.
πcos πx
1
views
Textbook Question
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
i. y = x² − 4
Textbook Question
Identifying Extrema
In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).
" style="" width="190">
a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.
1
views
Textbook Question
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
g(x) = x² − 4x + 4, 1 ≤ x < ∞