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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) = cos 3x, [π/12, 7π/12]
To verify that a function satisfies the three hypotheses of Rolle's Theorem, we need to check the following:
1. Continuity: The function must be continuous on the closed interval [a, b], where a and b are the endpoints of the interval.
2. Differentiability: The function must be differentiable on the open interval (a, b), meaning it has a derivative for every point in that interval.
3. Equal Function Values: The function values at the endpoints of the interval must be equal, i.e., f(a) = f(b).
Let's go through these three hypotheses for the given function f(x) = cos(3x) on the interval [π/12, 7π/12].
1. Continuity:
The function cos(3x) is continuous on its entire domain, which includes the interval [π/12, 7π/12]. Therefore, it satisfies the continuity hypothesis.
2. Differentiability:
The derivative of cos(3x) is -3sin(3x), which exists for all values of x. Hence, f(x) = cos(3x) is differentiable on the open interval (π/12, 7π/12). Therefore, it satisfies the differentiability hypothesis.
3. Equal Function Values:
Let's evaluate the function at the endpoints of the interval:
f(π/12) = cos(3(π/12)) = cos(π/4) = (sqrt(2))/2
f(7π/12) = cos(3(7π/12)) = cos(7π/4) = -(sqrt(2))/2
Since f(π/12) is not equal to f(7π/12), the function does not satisfy the equal function values hypothesis.
Therefore, the function f(x) = cos(3x) only satisfies two out of the three hypotheses of Rolle's Theorem on the interval [π/12, 7π/12].
Now, let's find all the numbers c that satisfy the conclusion of Rolle's Theorem. Rolle's Theorem states that if a function satisfies the three hypotheses:
1. Continuity on the closed interval [a, b].
2. Differentiability on the open interval (a, b).
3. Equal function values at the endpoints, f(a) = f(b).
Then there exists at least one number c in the open interval (a, b) such that f'(c) = 0.
Since the function does not satisfy the equal function values hypothesis, we cannot apply the conclusion of Rolle's Theorem in this case.