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.