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Chapter 2: Problem 38

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$f(x)=x^{5}-x^{3}-1 ; \text { between } 1 \text { and } 2$$

Short Answer

Expert verified
By the Intermediate Value Theorem, there exists at least one real zero for the given polynomial \(f(x) = x^{5} - x^{3} - 1\) between 1 and 2.

Step by step solution

01

Calculate \(f(1)\)

Replace \(x\) in \(f(x)\) with the first endpoint, 1: \(f(1) = (1)^{5} - (1)^{3} - 1 = 1 - 1 - 1 = -1\).
02

Calculate \(f(2)\)

Replace \(x\) in \(f(x)\) with the second endpoint, 2: \(f(2) = (2)^{5} - (2)^{3} - 1 = 32 - 8 - 1 = 23\).
03

Analyze the signs of \(f(1)\) and \(f(2)\)

The value of \(f(1)\) is -1, which is less than 0, and the value of \(f(2)\) is 23, which is greater than 0. Thus, the function \(f(x)\) takes on different signs at the endpoints of the interval [1, 2].
04

Apply the Intermediate Value Theorem

Since the function \(f(x)\) is continuous on the interval [1, 2] and takes on different signs at the endpoints of the interval, the Intermediate Value Theorem guarantees that there is at least one real zero in the interval (1, 2).

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