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\begin{tabular}{l|l} DETAILS & SCALC9 3.2.009.MI. \end{tabular}
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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.)
\[
f(x)=3 x^{2}-6 x+8,[-1,3]
\]
\[
c=
\]
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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 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]. We need to check if \(f(-1) = f(3)\) to satisfy the third condition of Rolle's theorem.

Step 3 ：If \(f(-1) = f(3)\), then the function satisfies all the conditions of Rolle's theorem. We can then find the value of c by setting the derivative of the function to zero and solving for x.

Step 4 ：The derivative of the function is \(f'(x) = 6x - 6\). Setting this equal to zero gives \(x = 1\).

Step 5 ：Final Answer: The function \(f(x)=3 x^{2}-6 x+8\) satisfies the conditions of Rolle's theorem on the interval \([-1,3]\). The value of \(c\) that satisfies the conclusion of Rolle's theorem is \(c=1\). So, \(c=\boxed{1}\).