Problem
Save \& Exit Practice Lesson: 6.3 Locating Real Zeros of Polyno... Question 5 of 10, Step 1 of 1 $4 / 15$ 3 Correct Find the value of the following function at $\mathrm{x}=8$ and $\mathrm{x}=9$. 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
Solution
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 ：\(\boxed{\text{There is a real zero between 8 and 9.}}\)
