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 4.3.39
Capitolo 4, Problema 4.3.39

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


f(x) = -12x⁵ + 75x⁴ - 80x³

Guida verificata passo dopo passo
1
To determine where the function f(x) = -12x⁵ + 75x⁴ - 80x³ is increasing or decreasing, we first need to find its derivative, f'(x). This will help us identify the critical points and analyze the behavior of the function.
Calculate the derivative f'(x) using the power rule. The derivative of f(x) = -12x⁵ + 75x⁴ - 80x³ is f'(x) = -60x⁴ + 300x³ - 240x².
Find the critical points by setting the derivative f'(x) equal to zero and solving for x. This involves solving the equation -60x⁴ + 300x³ - 240x² = 0.
Factor the equation -60x⁴ + 300x³ - 240x² = 0 to find the values of x that make the derivative zero. This can be done by factoring out the greatest common factor and solving the resulting polynomial equation.
Once the critical points are found, use a sign chart or test values in the intervals determined by these critical points to determine where f'(x) is positive (indicating f is increasing) and where f'(x) is negative (indicating f is decreasing).

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:
5m

Concetti chiave

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

Critical Points

Critical points are values of x where the derivative of a function is either zero or undefined. These points are essential for determining where a function changes from increasing to decreasing or vice versa. To find critical points, we first compute the derivative of the function and set it equal to zero, solving for x.
Video consigliato:
04:50
Critical Points

First Derivative Test

The First Derivative Test is a method used to determine the behavior of a function at its critical points. By analyzing the sign of the derivative before and after each critical point, we can conclude whether the function is increasing or decreasing in the intervals defined by these points. If the derivative changes from positive to negative, the function has a local maximum; if it changes from negative to positive, it has a local minimum.
Video consigliato:
07:09
The First Derivative Test: Finding Local Extrema

Intervals of Increase and Decrease

Intervals of increase and decrease refer to the ranges of x-values where a function is respectively rising or falling. A function is increasing on an interval if its derivative is positive throughout that interval, and it is decreasing if the derivative is negative. Identifying these intervals helps in understanding the overall behavior of the function and is crucial for graphing and optimization problems.
Video consigliato:
07:32
Determining Where a Function is Increasing & Decreasing
Pratica correlata
Domanda del libro di testo

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


f(x) = √2 sin x- x on [0, 2π]

253
views
Domanda del libro di testo

Linear approximation Find the linear approximation to the following functions at the given point a.


g(t) = √(2t + 9); a = -4

230
views
Domanda del libro di testo

Sketch the graph of a continuous function ƒ on [0, 4] satisfying the given properties.


ƒ' (x) = 0 for x = 1 and 2; ƒ has an absolute maximum at x = 4; ƒ has an absolute minimum at x= 0; and ƒ has a local minimum at x = 2.

280
views
Domanda del libro di testo

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (3x + 1) (4 - x) dx

83
views
Domanda del libro di testo

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


lim_x→ 0 (eˣ - 1) / (x² + 3x)

272
views
Domanda del libro di testo

{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.


f(x) = tan x - 2x; x₀ = 1.2

263
views