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DETAILS SCALC9 3.2.009.MI.
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Verify that the function satisfies the three hypotheses of Rolle's theorem on the given interval. Then find all numbers $c$ that satisfy the conclusion of Rolle's theorem. (Enter your answers as a comma-separated list.)
\[
f(x)=3 x^{2}-6 x+8, \quad[-1,3]
\]
\[
c=
\]
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Step 1 ：Rolle's Theorem states that if a function f is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one number c in the open interval (a, b) such that the derivative of f at c is zero.

Step 2 ：In this case, the function \(f(x) = 3x^2 - 6x + 8\) is a polynomial, so it is continuous and differentiable on all real numbers, including the interval [-1, 3].

Step 3 ：We need to check if \(f(-1) = f(3)\). If this is true, then by Rolle's Theorem, there exists at least one number c in the interval (-1, 3) such that the derivative of f at c is zero.

Step 4 ：Calculating \(f(-1)\) and \(f(3)\), we find that both are equal to 17. This means the function satisfies the condition \(f(a) = f(b)\) of Rolle's Theorem.

Step 5 ：The next step is to find the derivative of the function and set it equal to zero to find the value(s) of c in the interval (-1, 3) that satisfy the conclusion of Rolle's Theorem.

Step 6 ：The derivative of the function is \(6x - 6\), and the solution to the equation \(6x - 6 = 0\) is \(x = 1\).

Step 7 ：This means that there is one number c in the interval (-1, 3) such that the derivative of f at c is zero, and that number is 1.

Step 8 ：Final Answer: The number \(c\) that satisfies the conclusion of Rolle's theorem is \(\boxed{1}\).