[tex]\begin{gathered} f(x)=\cot (\frac{x}{2}) \\ \lbrack\pi,9\pi\rbrack \end{gathered}[/tex]
Let see what Rolle's theorem mean:
Let f(x) be a function that satisfies the following 3 hypotheses:
1. f(x) is continuous on the closed interval [a,b]
2. f(x) is differentiable on the open interval (a,b)
3. f(a) = f(b)
Then there is a number c in (a,b) such that f'(c) = 0
Let's check every condition:
1.
As we can see, the function is not continuous for:
[tex]2\pi,4\pi,6\pi,8\pi[/tex]
2.
Since the function is not continuous for some points over the interval, we can also conclude the function is not differentiable for (a,b).
3.
We can see that f(a) = f(b).
Therefore, the answers are:
There are points on the interval (a,b) where f is not diffenrentiable
There are points on the interval [a,b] where f is not continuous
