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.) \[ f(x)=3 x^{2}-6 x+2, \quad[-1,3] \] \[ c= \]

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 ：We need to check if the function \(f(x)=3 x^{2}-6 x+2\) satisfies these three conditions on the interval [-1, 3].

Step 3 ：The function is a polynomial, so it is continuous and differentiable on the entire real line, including the interval [-1, 3].

Step 4 ：We need to check if \(f(-1) = f(3)\).

Step 5 ：If the function satisfies these conditions, then we can find the derivative of the function and set it equal to zero to find the values of c that satisfy the conclusion of Rolle's theorem.

Step 6 ：The function does satisfy the conditions of Rolle's theorem on the interval [-1, 3] because it is continuous and differentiable on this interval and \(f(-1) = f(3)\).

Step 7 ：The value of c that satisfies the conclusion of Rolle's theorem is 1, because the derivative of the function is zero at this point.

Step 8 ：Final Answer: The numbers that satisfy the conclusion of Rolle's theorem are \(c = \boxed{1}\).