Explain why rolle's theorem does not apply to the function even though there exist a and b such that f(a) = f(b).
f(x) = cot (x/2) , [π, 9π]

This may help you

he functión Cot[x/2] is not continuos in the points x=2nπ,  where  n=0,1,2,3,... You can check it with a calculator. So the function is not continuos in the domain the problem gives you, so the Rolle's theorem can not be applied. If the inteverval was, [π/2,3π/2] Then we could apply the Rolle's theorem.