[Technology Exercise] Roots

Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.

a. Use the Intermediate Value Theorem to show that ƒ has a zero between ―1 and 2 .

검증된 단계별 안내

First, understand the Intermediate Value Theorem (IVT): It states that if a function f is continuous on a closed interval [a, b] and N is any number between f(a) and f(b), then there exists at least one c in the interval (a, b) such that f(c) = N.

Check the continuity of the function f(𝓍) = 𝓍³ - 𝓍 - 1. Since this is a polynomial function, it is continuous everywhere, including on the interval [-1, 2].

Evaluate the function at the endpoints of the interval: Calculate f(-1) and f(2).

Calculate f(-1): Substitute 𝓍 = -1 into the function: f(-1) = (-1)³ - (-1) - 1. Simplify this expression to find the value of f(-1).

Calculate f(2): Substitute 𝓍 = 2 into the function: f(2) = (2)³ - (2) - 1. Simplify this expression to find the value of f(2).

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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 on different signs at the endpoints, then there exists at least one c in (a, b) such that f(c) = 0. This theorem is crucial for proving the existence of roots within a specified interval.

Continuity of Functions

A function is continuous if there are no breaks, jumps, or holes in its graph. For the Intermediate Value Theorem to apply, the function must be continuous over the interval in question. Polynomial functions, like ƒ(𝓍) = 𝓍³ - 𝓍 - 1, are continuous everywhere, making them suitable for this theorem.

Evaluating Function Values

To apply the Intermediate Value Theorem, we need to evaluate the function at the endpoints of the interval. By calculating ƒ(-1) and ƒ(2), we can determine the signs of these values. If they are of opposite signs, it confirms the existence of at least one root in the interval (-1, 2).

[Technology Exercise] RootsLet ƒ(𝓍) = 𝓍³ ―𝓍― 1.a. Use the Intermediate Value Theorem to show that ƒ has a zero between ―1 and 2 .