Let's go through each part of the problem step by step.

Part a: Apply Rolle's Theorem to $f(x) = \sin(x) \sqrt{\cos(2x)}$ on the interval $(0, \pi)$.

Rolle's Theorem states that if a function $f(x)$ is continuous on the closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c$ in the interval $(a, b)$ such that $f'(c) = 0$.

Step 1: Check the conditions of Rolle's Theorem.

1. Continuity: The function ( f(x) = \sin(x) \sqrt{\cos(2x)} \