To verify the hypotheses of Rolle's Theorem and find the value(s) of $c$, follow these steps:

Step 1: Verify the hypotheses of Rolle's Theorem

1. The function $f(x) = \sin(x/2)$ is continuous on $[ \pi/2, 3\pi/2 ]$.

   • $\sin(x/2)$ is a sine function, which is continuous everywhere. Hence, it is continuous on $[ \pi/2, 3\pi/2 ]$.

2. The function $f(x) = \sin(x/2)$ is differentiable on $( \pi/2, 3\pi/2 )$.

   • $\sin(x/2)$ is differentiable everywhere because it is a sine function, and its derivative $f'(x) = (1/2)\cos(x/2)$ exists for all $x$.

3. $f(\pi/2) = f(3\pi/2)$: The function has equal values at the endpoints.

   • At $x = \pi/2$: $f(\pi/2) = \sin((\pi/2)/2) = \sin(\pi/4) = \frac{\sqrt{2}}{2}.$
   • At $x = 3\pi/2$: $f(3\pi/2) = \sin((3\pi/2)/2) = \sin(3\pi/4) = \frac{\sqrt{2}}{2}.$
   • Since $f(\pi/2) = f(3\pi/2)$, the third hypothesis is satisfied.

Thus, all three hypotheses of Rolle's Theorem are satisfied.

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Step 2: Apply Rolle's Theorem

Rolle's Theorem states that there exists at least one $c$ in $(\pi/2, 3\pi/2)$ such that $f'(c) = 0$.

1. Find $f'(x)$: $f(x) = \sin(x/2), \quad f'(x) = \frac{1}{2} \cos(x/2).$

2. Set $f'(x) = 0$: $\frac{1}{2} \cos(x/2) = 0 \implies \cos(x/2) = 0.$

3. Solve for $x$: $\cos(x/2) = 0 \implies x/2 = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}.$ Multiply through by 2: $x = \pi + 2k\pi, \quad k \in \mathbb{Z}.$

4. Find $x$ in the interval $(\pi/2, 3\pi/2)$:

   • For $k = 0$, $x = \pi$.
   • $x = \pi$ lies in $(\pi/2, 3\pi/2)$.

Thus, the only value of $c$ is: $c = \pi.$

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Final Answer:

The value of $c$ that satisfies the conclusion of Rolle's Theorem is: $\boxed{\pi}.$

Do you want detailed explanations of any of the steps or further elaboration?

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Expanding Questions:

1. How does differentiability affect the applicability of Rolle's Theorem?
2. Can a discontinuous function satisfy Rolle's Theorem on a given interval? Why or why not?
3. What is the geometric interpretation of Rolle's Theorem?
4. What would happen if the endpoints of the function were not equal?
5. Can you generalize Rolle’s Theorem to higher-dimensional functions?

Tip: When verifying Rolle's Theorem, always check each hypothesis carefully; even one failure invalidates the theorem!