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Chapter 2: Problem 40

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$f(x)=3 x^{3}-8 x^{2}+x+2 ; \text { between } 2 \text { and } 3$$

Short Answer

Expert verified
By utilizing the Intermediate Value Theorem, it is confirmed that there exists at least one real zero for the function \(f(x) = 3x^{3} - 8x^{2} + x + 2\) between the values 2 and 3.

Step by step solution

01

Calculate values of function at the endpoints

Calculate the function value at 2 and 3. That yields \(f(2) = 3*2^{3} - 8*2^{2} + 2 + 2 = -4\) and \(f(3) = 3*3^{3} - 8*3^{2} + 3 + 2 = 10\).
02

Apply the Intermediate Value Theorem

The function \(f(x)\) is a polynomial function, which means it is continuous at every real number. Here \(f(2) < 0\) and \(f(3) > 0\), and since 0 is between these function values, by the Intermediate Value Theorem, there must be at least one value \(c\) for \(2 < c < 3\) such that \(f(c) = 0\).
03

Conclusion

By using the Intermediate Value Theorem (IVT), it has been shown that there is at least one real zero between 2 and 3 for the function \(f(x) = 3x^{3} - 8x^{2} + x + 2\).

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