Explain why Rolle's Theorem does not apply to the function even though there exist a and b such that f(a) = f(b). (Select all that apply.) f(x) = cot x/2, [pi, 5pi]

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12-06-2022
Mathematics

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Explain why Rolle's Theorem does not apply to the function even though there exist a and b such that f(a) = f(b). (Select all that apply.) f(x) = cot x/2, [pi, 5pi]

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maheshpatelvVT maheshpatelvVT · Answer 1 · 2022-06-18T06:11:48+02:00

There are points on the interval [π, 5π] where f is not continuous and there are points on the interval (π, 5π) where f is not differentiable.

What is Rolle's theorem?

Rolle's theorem, sometimes known as Rolle's lemma, states that any real-valued differentiable function that reaches equal values at two independent places must have at least one stationary point in between, i.e. a point where the first derivative is zero.

As we know, the condition for the Rolle's theorem is:

f(a) = f(b)
f(x) must be continuous in [a, b]
f(x) must be differentiable in (a, b)

The function is:

f(x) = cotx/2

The cot function is not continuous at x = nπ, n ≠ 0

The cot functions are not differentiable at x = nπ, n ≠ 0

Thus, there are points on the interval [π, 5π] where f is not continuous and there are points on the interval (π, 5π) where f is not differentiable.

Learn more about the Rolle's theorem here:

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