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.)

Question

f(x)=cotx/2 [pi,3pi]

a)There are points on the interval (a, b) where f is not differentiable.

b)None of these

c)f'(a) does not equal f(b) for all possible values of a and b in the interval [π, 3π].

d)f ''(a) does not equal f '(b) for any values in the interval [π, 3π].

e) There are points on the interval [a, b] where f is not continuous.

Answer 1

Rolle's Theorem has the following conditions:

1. f(x) must be continuous in [a,b]
2. f(x) must be differentiable in (a,b)
3. f(a) = f(b)

Now ask yourself the following questions.

1. Is f(pi) = f(3pi)?
2. Is f(x) = cot(x/2) continuous at all points in [pi,3pi]? You may want to use a graph of the function to see this.
3. If the function is not continuous at a point or points within (pi,3pi), then it is also not differentiable at these points. Note that the reverse is not necessarily true, i.e. a function can be continuous at all points within [a,b], but not necessarily differentiable at all points within (a,b). Is there a point or points within (pi,3pi) where the function is not differentiable?

Obviously, if you can answer 2, then you can answer 3.

If any of the three conditions fail, then Rolle's Theorem does not apply.