Question

use the intermediate value theorem to show that f has a zero between a and b. 2) f(x)=-x^{4}+3x^{3}-2x + 1 a = 2, b = 3

Explanation:

Step1: Evaluate \(f(a)\)

Substitute \(x = 2\) into \(f(x)=-x^{4}+3x^{3}-2x + 1\).

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Step2: Evaluate \(f(b)\)

Substitute \(x = 3\) into \(f(x)=-x^{4}+3x^{3}-2x + 1\).

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Step3: Apply the Intermediate Value Theorem

Since \(f(x)\) is a polynomial, it is continuous on the interval \([2,3]\).
We have \(f(2)=5>0\) and \(f(3)=-5 < 0\).
By the Intermediate Value Theorem, for \(N = 0\) (since we want to find a zero), there exists a \(c\in(2,3)\) such that \(f(c)=0\).

Answer:

Since \(f(2)=5>0\) and \(f(3)=-5<0\) and \(f(x)\) is continuous on \([2,3]\), by the Intermediate Value Theorem, \(f(x)\) has a zero between \(2\) and \(3\).