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(5x), [π/20, 7π/20] c =

There are 2 points where f'(x) = 0

At x = 0, and at x = 5/π

Step-by-step explanation:

Rolle's theorem states that for any differentiable function on the interval {a, b}, if f(a) = f(b), then there is at least one point in the interval where f'(c) = 0

Evaluate the end points of the interval to see if we can apply Rolle's Theorem...

f(π/20) = cos (5π/20) = cos (π/4) = (√2)/2

f(7π/20) = cos [5(7π]/20) = cos (35π/20) = cos (7π/4) = (√2)/2

So by Rolle's Theorem, there will be at least one point where f'(c) = 0, so find f'(x)

f'(x) = -5sin(5x)

find where this equal zero...

0 = -5sin(5x)

0 = sin(5x)

Sin x = 0 at x = 0, and x = π, so we have..

5x = 0,  so x = 0

5x = π, so x = 5/π

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