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Chapter 2: Problem 36

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$f(x)=x^{4}+6 x^{3}-18 x^{2} ; \text { between } 2 \text { and } 3$$

Short Answer

Expert verified
By using the Intermediate Value Theorem, it can be shown that the polynomial \(f(x) = x^{4}+6x^{3}-18x^{2}\) has a real root in the interval between 2 and 3.

Step by step solution

01

Finding f(2)

Substitute \(x = 2\) into the polynomial function. Therefore, \(f(2) = (2)^{4}+6(2)^{3}-18(2)^{2} = 16 + 48 - 72 = -8.\)
02

Finding f(3)

Next, substitute \(x = 3\) into the polynomial function. Therefore, \(f(3) = (3)^{4}+6(3)^{3}-18(3)^{2} = 81 + 162 - 162 = 81.\)
03

Evaluating Signs

We observe that \(f(2) = -8\) and \(f(3) = 81\). Therefore, \(f(2)\) and \(f(3)\) have different signs.
04

Applying the Intermediate Value Theorem

Since the function \(f(x) = x^{4}+6x^{3}-18x^{2}\) is a polynomial, it is continuous everywhere. Also, because \(f(2)\) and \(f(3)\) have different signs, by the Intermediate Value Theorem, there exists a number \(c\) in the interval (2,3) such that \(f(c) = 0\). Therefore, the polynomial has a real zero between 2 and 3.

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