Find the value c that satisfies Rolle's Theorem for f(x) = 9 cos 3x on .
C) 0

Question

Find the value c that satisfies Rolle's Theorem for f(x) = 9 cos 3x on  .
C) 0

Quizplus · Accepted Answer

Rolle's Theorem states that if a function is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), and if f(a) = f(b), then there exists at least one c in (a,b) such that f'(c) = 0. 
For f(x) = 9cos(3x), f(a) = f(b) since cos(3x) is a periodic function with a period of 2pi/3. Therefore, we can choose any interval with length equal to a multiple of 2pi/3, such as [0,2pi/3] or [-pi/3, pi/3]. 
Since f(x) is differentiable for all x, the only possible value of c that satisfies Rolle's Theorem is when f'(c) = 0. Taking the derivative of f(x), we have f'(x) = -27sin(3x). Setting f'(c) = 0, we get -27sin(3c) = 0, which implies sin(3c) = 0. The solutions to sin(3c) = 0 are c = n(pi/3), where n is an integer. Therefore, we can choose c = 0, pi/3, 2pi/3, etc. 
Out of the given answer choices, the only option that includes c = 0 is choice C, which is the correct answer.

Find the Value C That Satisfies Rolle's Theorem for F(x)