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EXAMPLE 10 Show that there is a root of the equation 2x3 − 4x2 + 3x − 2 = 0 between 1 and 2. SOLUTION Let f(x) = 2x3 − 4x2 + 3x − 2 = 0. We are looking for a solution of the given equation, that is, a number c between 1 and 2 such that f(c) = . Therefore we take a = , b = , and N = in the Intermediate Value Theorem. We have

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Answer: [tex]c = 1.2[/tex]

Step-by-step explanation:

First, it is needed to determined the values for x = 1 and x = 2:

[tex]f(1) = -1, f(2)=4[/tex]

The sign change within the interval is the most sound evidence of the root existence. According to the Intermediate Value Theorem, there is a number [tex]c[/tex] such that [tex]f(c) = 0[/tex]. Another finding is that [tex]c[/tex] is closer to 1 than to 2.

[tex]c = a + \frac{b-a}{f(b)-f(a)}\cdot[f(c)-f(a)][/tex]

[tex]c = 1 + \frac{2-1}{4-(-1)}\cdot[0-(-1)] \\c = 1.2[/tex]

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