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Chapter 2: Problem 33

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

Short Answer

Expert verified
By substituting the end points of the interval into the equation, and then applying the Intermediate Value Theorem, we can confirm that there exists a real zero in the interval (1,2) for the function \(f(x) = x^{3} -x -1\).

Step by step solution

01

Evaluate the function at the given interval end points

First, substitute the given points \(1\) and \(2\) into the function. Thus, we will have: \(f(1) = (1)^{3} - (1) - 1 = -1\) \(f(2) = (2)^{3} - (2) - 1 = 5\)
02

Apply the Intermediate Value Theorem

From step 1, we have \(f(1) < 0\) and \(f(2) > 0\). Since \(f(x)\) is a polynomial (and thus continuous), the IVT ensures that at some point \(c\) in (1, 2), \(f(c) = 0\). Hence, there exists a real zero of the polynomial between 1 and 2.

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