find the value of c that satisfies rolles theorem on [0,2pi] for the function sin2x

To find the value(s) of \( c \) that satisfy Rolle's Theorem for \( f(x) = \sin(2x) \) on the interval \([0, 2\pi]\), we proceed as follows:

### Steps:
1. **Check the conditions of Rolle's Theorem:**
   - The function \( f(x) \) is continuous on \([0, 2\pi]\) and differentiable on \((0, 2\pi]\).
   - \( f(0) = \sin(2 \cdot 0) = 0 \) and \( f(2\pi) = \sin(2 \cdot 2\pi) = 0 \), so \( f(0) = f(2\pi) \).

   Since \( f(x) \) satisfies all conditions of Rolle's Theorem, there exists at least one \( c \in (0, 2\pi) \) such that \( f'(c) = 0 \).

2. **Find the derivative \( f'(x) \):**
   \[
   f'(x) = \frac{d}{dx} \sin(2x) = 2\cos(2x).
   \]

3. **Set \( f'(c) = 0 \) and solve:**
   \[
   2\cos(2c) = 0 \quad \Rightarrow \quad \cos(2c) = 0.
   \]

   The cosine function is zero at \( 2c = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \dots \), which simplifies to:
   \[
   2c = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}.
   \]

   Dividing by 2:
   \[
   c = \frac{\pi}{4} + \frac{n\pi}{2}, \quad n \in \mathbb{Z}.
   \]

4. **Find \( c \) within the interval \( (0, 2\pi) \):**
   Substitute \( n = 0, 1, 2, 3, \dots \) and keep \( c \) in \( (0, 2\pi) \):
   - \( n = 0 \): \( c = \frac{\pi}{4} \).
   - \( n = 1 \): \( c = \frac{\pi}{4} + \frac{\pi}{2} = \frac{3\pi}{4} \).
   - \( n = 2 \): \( c = \frac{\pi}{4} + \pi = \frac{5\pi}{4} \).
   - \( n = 3 \): \( c = \frac{\pi}{4} + \frac{3\pi}{2} = \frac{7\pi}{4} \).
   - \( n = 4 \): \( c = \frac{\pi}{4} + 2\pi = \frac{9\pi}{4} \), which is outside \((0, 2\pi)\).

Thus, the values of \( c \) are:
\[
c = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}.
\]

### Final Answer:
The values of \( c \) that satisfy Rolle's Theorem are:
\[
c = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}.
\]

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### Related Questions:
1. How does differentiability ensure the existence of a critical point in Rolle's Theorem?  
2. What is the geometrical interpretation of Rolle's Theorem for \( \sin(2x) \)?  
3. What happens if \( f(a) \neq f(b) \) in Rolle’s Theorem?  
4. How does \( \cos(2x) = 0 \) relate to the critical points of \( \sin(2x) \)?  
5. Can Rolle's Theorem be applied to non-polynomial functions, and why?

### Tip:
To apply Rolle's Theorem, always verify the conditions (continuity, differentiability, and equal endpoints) before solving.

Find the value of c that satisfies Rolle's Theorem on [0, 2π] for the function sin(2x).

Discover how to apply Rolle's Theorem to the function sin(2x) on the interval [0, 2π]. This detailed guide verifies the conditions of Rolle's Theorem and calculates critical points where the derivative equals zero. Perfect for high school calculus students in Grades 11-12.

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