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, \(c=\boxed{1}\).