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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Lial 13th EditionCh. 3 - Polynomial and Rational Functions문제 48
4장, 문제 48

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. ƒ(x)=3x2-x-4; 1 and 2

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Recall the Intermediate Value Theorem (IVT), which states that if a function \( f(x) \) 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: \( f(x) = 3x^2 - x - 4 \) and the interval \([1, 2]\).
Evaluate \( f(1) \) by substituting \( x = 1 \) into the function: \( f(1) = 3(1)^2 - 1 - 4 \).
Evaluate \( f(2) \) by substituting \( x = 2 \) into the function: \( f(2) = 3(2)^2 - 2 - 4 \).
Check the signs of \( f(1) \) and \( f(2) \). If one is positive and the other is negative, then by the IVT, there is at least one real zero of \( f(x) \) between 1 and 2.

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주요 개념

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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.
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6:15
Introduction to Hyperbolas

Polynomial Continuity

Polynomial functions are continuous everywhere on the real number line, meaning there are no breaks, jumps, or holes in their graphs. This continuity ensures that the Intermediate Value Theorem can be applied to any interval when analyzing polynomial functions.
추천 영상:
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Introduction to Polynomial Functions

Evaluating Function Values at Interval Endpoints

To apply the Intermediate Value Theorem, you calculate the function's values at the endpoints of the interval. If these values have opposite signs, it indicates the function crosses zero somewhere between, confirming the existence of a real root in that interval.
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