교과서 질문
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. (2−x)2(x−7/2)<0
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
4장, 문제 34
Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x3−4x2+2; between 0 and 1
검증된 단계별 안내1
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, \(f(x) = x^{3} - 4x^{2} + 2\) and the interval is \([0, 1]\).
Evaluate the function at the endpoints of the interval: calculate \(f(0)\) and \(f(1)\).
Check the signs of \(f(0)\) and \(f(1)\): if one is positive and the other is negative, then by the IVT, there is at least one root between 0 and 1.
Conclude that since \(f\) is a polynomial (and thus continuous everywhere) and the function values at 0 and 1 have opposite signs, there must be a real zero of \(f(x)\) between 0 and 1.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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 takes any value between f(a) and f(b) at some point within the interval. This theorem is used to prove the existence of roots by showing the function changes sign.
추천 영상:
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 any interval when dealing with polynomials.
추천 영상:
06:04Introduction to Polynomial Functions
Evaluating Function Values at Given Points
To apply the Intermediate Value Theorem, you must calculate the function's values at the endpoints of the interval. If the function values have opposite signs, it indicates the function crosses zero somewhere between those points, confirming the existence of a real root.
추천 영상:
4:26Evaluating Composed Functions
관련 실천
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Use the Rational Zero Theorem to list all possible rational zeros for each given function.
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