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) = 3x2 − 6x + 4, [−1, 3]
Question
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) = 3x2 − 6x + 4, [−1, 3]
Solution
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 interval (a, b) such that the derivative of f at c is zero.
Let's verify these conditions for the function f(x) = 3x^2 - 6x + 4 on the interval [-1, 3].
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Continuity: The function f(x) = 3x^2 - 6x + 4 is a polynomial function, which is continuous everywhere, including the interval [-1, 3].
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Differentiability: The function f(x) = 3x^2 - 6x + 4 is differentiable everywhere because it's a polynomial, so it's differentiable on the open interval (-1, 3).
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f(a) = f(b): We need to check if f(-1) = f(3).
f(-1) = 3(-1)^2 - 6(-1) + 4 = 3 + 6 + 4 = 13 f(3) = 3(3)^2 - 6(3) + 4 = 27 - 18 + 4 = 13
So, f(-1) = f(3).
Since all three conditions are satisfied, Rolle's Theorem applies.
Now, we need to find the value of c in the interval (-1, 3) such that f'(c) = 0.
The derivative of f(x) = 3x^2 - 6x + 4 is f'(x) = 6x - 6.
Setting f'(x) = 0 gives 6x - 6 = 0, which simplifies to x = 1.
So, the number c that satisfies the conclusion of Rolle's Theorem is c = 1.
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