Verify Rolle's theorem for f (x)=sinx+2 for 0=<x=<2π

To verify Rolle's Theorem for \( f(x) = \sin(x) + 2 \) on the interval \( [0, 2\pi] \), we need to ensure that the function satisfies the three conditions of Rolle's Theorem:

### Rolle's Theorem Conditions:
1. **The function must be continuous on the closed interval** \([a, b]\).
2. **The function must be differentiable on the open interval** \((a, b)\).
3. **The function values at the endpoints must be equal**, i.e., \( f(a) = f(b) \).

### Step 1: Continuity and Differentiability
- The function \( f(x) = \sin(x) + 2 \) is continuous and differentiable everywhere, including on the interval \([0, 2\pi]\), because sine is continuous and differentiable, and adding a constant does not affect continuity or differentiability.
  
Thus, the first two conditions are satisfied.

### Step 2: Checking \( f(a) = f(b) \)
We now check the values of the function at the endpoints \( x = 0 \) and \( x = 2\pi \).

- \( f(0) = \sin(0) + 2 = 0 + 2 = 2 \)
- \( f(2\pi) = \sin(2\pi) + 2 = 0 + 2 = 2 \)

Since \( f(0) = f(2\pi) \), the third condition is satisfied.

### Step 3: Finding \( c \in (0, 2\pi) \) such that \( f'(c) = 0 \)
Now, we need to find a point \( c \in (0, 2\pi) \) where the derivative of \( f(x) \) equals zero.

- First, compute the derivative of \( f(x) \):
  \[
  f'(x) = \cos(x)
  \]

- Now, solve for \( x \) such that \( f'(x) = 0 \):
  \[
  \cos(x) = 0
  \]
  The cosine function equals zero at:
  \[
  x = \frac{\pi}{2}, \frac{3\pi}{2}
  \]

Both \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \) lie within the open interval \( (0, 2\pi) \).

### Conclusion:
Rolle's Theorem is satisfied for \( f(x) = \sin(x) + 2 \) on the interval \( [0, 2\pi] \), with points \( c = \frac{\pi}{2} \) and \( c = \frac{3\pi}{2} \) where \( f'(c) = 0 \).

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Would you like further clarification on any step? Here are five related questions to expand your understanding:

1. How does Rolle's Theorem apply to non-trigonometric functions?
2. What happens if the function isn't differentiable at some point within the interval?
3. How can we interpret the geometric meaning of Rolle's Theorem?
4. Can a function have more than one value of \( c \) that satisfies Rolle's Theorem?
5. How does Rolle’s Theorem relate to the Mean Value Theorem?

**Tip:** When solving Rolle’s Theorem problems, always verify continuity, differentiability, and endpoint equality before finding where the derivative is zero.

Verify Rolle's theorem for f(x) = sin(x) + 2 for 0 ≤ x ≤ 2π

Explore how to verify Rolle's Theorem for the function f(x) = sin(x) + 2 on the interval [0, 2π]. This step-by-step solution covers the continuity, differentiability, and endpoint conditions, with solutions for c values where f'(x) = 0.

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