Save \& Exit Practice Lesson: 6.3 Locating Real Zeros of Polyno...

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Find the value of the following function at $x = 8$ and $x = 9$ . Does the Intermediate Value Theorem guarantee that the function has a real zero between these two $x$ values?
$f (x) = x^{3} - 9 x^{2} + 2 x - 5$

Answer
$\begin{array}{l} f (8) = \\ f (9) = \end{array}$

Does the intermediate Value Theorem guarantee that there is a real zero between 8 and 9 ?
YES ONO

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Answer

$There is a real zero between 8 and 9.$

Steps

Step 1 ：Calculate the value of the function at $x = 8$ : $f (8) = 8^{3} - 9 * 8^{2} + 2 * 8 - 5 = 512 - 576 + 16 - 5 = - 53$

Step 2 ：Calculate the value of the function at $x = 9$ : $f (9) = 9^{3} - 9 * 9^{2} + 2 * 9 - 5 = 729 - 729 + 18 - 5 = 13$

Step 3 ：Since $f (8) = - 53$ and $f (9) = 13$ , and 0 is between -53 and 13, the Intermediate Value Theorem guarantees that there is a real zero between 8 and 9.

Step 4 ： $There is a real zero between 8 and 9.$

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