Skip to main content
Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 38
Capitolo 4, Problema 38

In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x5−x3−1; between 1 and 2

Guida verificata passo dopo passo
1
Recall the Intermediate Value Theorem (IVT), which states that if a function f is continuous on a closed interval [a, b] and f(a) and f(b) have opposite signs, then there exists at least one c in (a, b) such that f(c) = 0.
Identify the function and the interval: here, f(x) = x5 - x3 - 1, and the interval is [1, 2].
Evaluate f at the endpoints: calculate f(1) and f(2) by substituting x = 1 and x = 2 into the function without simplifying the final numeric value.
Check the signs of f(1) and f(2): determine whether f(1) and f(2) are positive or negative to see if they have opposite signs.
Since f is a polynomial (which is continuous everywhere) and f(1) and f(2) have opposite signs, conclude by the Intermediate Value Theorem that there is at least one real zero of f(x) between 1 and 2.

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.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes values f(a) and f(b) at each end, then it must take any value between f(a) and f(b) at some point within the interval. This theorem is used to prove the existence of roots within an interval.
Video consigliato:
6:15
Introduction to Hyperbolas

Continuity of Polynomial Functions

Polynomial functions are continuous everywhere on the real number line, meaning there are no breaks, jumps, or holes in their graphs. This property ensures that the Intermediate Value Theorem can be applied to polynomials on any interval.
Video consigliato:
06:04
Introduction to Polynomial Functions

Evaluating Function Values at Interval Endpoints

To apply the Intermediate Value Theorem, you calculate the function's values at the given interval endpoints. If the function values have opposite signs, it indicates the function crosses zero within the interval, confirming the existence of a real root.
Video consigliato:
4:26
Evaluating Composed Functions
Pratica correlata
Domanda del libro di testo

In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=2x−x2−2

1012
views
Domanda del libro di testo

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x3+x2+4x+4>0x^3+x^2+4x+4>0

507
views
Domanda del libro di testo

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x3−3x2−9x+27<0x^3−3x^2−9x+27<0

618
views
Domanda del libro di testo

Find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=12x/(3x2+1)

1051
views
Domanda del libro di testo

Find the horizontal asymptote, if there is one, of the graph of each rational function. g(x)=12x2/(3x2+1)

909
views
Domanda del libro di testo

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=3x3−10x+9; between -3 and -2

577
views