Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.4.14d
Capitolo 4, Problema 4.4.14d

if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.


d. Identify the local extreme values and inflection points of ƒ .

Guida verificata passo dopo passo
1
To find the local extreme values, we need to analyze the first derivative, ƒ'(x). Set ƒ'(x) = 0 to find critical points. This involves solving the equation 12x³ / (3x⁴ + 5)² = 0, which simplifies to finding when the numerator, 12x³, is zero.
The critical points occur when 12x³ = 0, which simplifies to x = 0. This is the only critical point since the denominator (3x⁴ + 5)² is never zero.
To determine if x = 0 is a local maximum or minimum, use the second derivative test. Evaluate ƒ''(x) at x = 0. If ƒ''(0) > 0, x = 0 is a local minimum; if ƒ''(0) < 0, x = 0 is a local maximum.
Next, identify the inflection points by setting the second derivative, ƒ''(x), equal to zero. Solve 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ = 0. This requires finding when the numerator, 180x² (x² + 1) (x + 1) (x - 1), is zero.
The potential inflection points occur when x² = 0, x² + 1 = 0, x + 1 = 0, or x - 1 = 0. Solve these equations to find x = 0, x = -1, and x = 1. Check the sign change of ƒ''(x) around these points to confirm they are inflection points.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
8m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Local Extreme Values

Local extreme values refer to the points in a function where it reaches a local maximum or minimum. These points occur where the first derivative, ƒ'(x), is equal to zero or undefined. To identify these points, one typically analyzes the sign changes of the first derivative around critical points, which helps determine whether the function is increasing or decreasing.
Video consigliato:
Percorso guidato
06:37
Average Value of a Function

Inflection Points

Inflection points are points on the graph of a function where the concavity changes, which can be identified by examining the second derivative, ƒ''(x). Specifically, an inflection point occurs where ƒ''(x) equals zero or is undefined, and the sign of ƒ''(x) changes around that point. This indicates a transition in the curvature of the graph, which is crucial for understanding the function's behavior.
Video consigliato:
04:50
Critical Points

First and Second Derivatives

The first derivative of a function, ƒ'(x), provides information about the function's rate of change and is essential for finding local extrema. The second derivative, ƒ''(x), offers insights into the function's concavity and helps identify inflection points. Together, these derivatives are fundamental tools in calculus for analyzing the behavior of functions and understanding their graphical representations.
Video consigliato:
06:02
The Second Derivative Test: Finding Local Extrema
Pratica correlata
Domanda del libro di testo

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

d. Find the time when the object strikes the ground.

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

56
views
Domanda del libro di testo

{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.


c. By experimentation, determine the approximate value of a (3 < a < 4)at which the graph separates into two curves.

263
views
Domanda del libro di testo

Rectangles in triangles Find the dimensions and area of the rectangle of maximum area that can be inscribed in the following figures.

d. An arbitrary triangle with a given area A (The result applies to any triangle, but first consider triangles for which all the angles are less than or equal to 90° .)

373
views
Domanda del libro di testo

Sketch a graph of a function f with the following properties.


f' < 0 and f" < 0, for 8 < x < 10

264
views
Domanda del libro di testo

{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51). 


c. Graph g for a = 2, 3, and 4.

220
views
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


c. If f(x) = mx + b, then the linear approximation to f at any point is L(x) = f(x).

186
views