Examine if Rolle’s theorem is applicable for f (x) = sec x in [0, 2π].

Rolle's Theorem states that if a function \( f \) is:
1. Continuous on the closed interval \([a, b]\),
2. Differentiable on the open interval \( (a, b) \),
3. Satisfies \( f(a) = f(b) \),

then there exists at least one point \( c \) in the open interval \( (a, b) \) such that \( f'(c) = 0 \).

To examine if Rolle's theorem is applicable to \( f(x) = \sec x \) on the interval \([0, 2\pi]\):

### Step 1: Check Continuity on \([0, 2\pi]\)
The function \( f(x) = \sec x = \frac{1}{\cos x} \) is continuous wherever \( \cos x \neq 0 \). However, \( \cos x = 0 \) at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \). Both of these points lie within the interval \([0, 2\pi]\), which means \( f(x) \) is **not continuous** on \([0, 2\pi]\).

### Step 2: Check Differentiability on \( (0, 2\pi) \)
\( f(x) = \sec x \) is differentiable wherever it is continuous. Since it is not continuous at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \), it is also not differentiable at these points within \( (0, 2\pi) \).

### Step 3: Check if \( f(0) = f(2\pi) \)
\[
f(0) = \sec(0) = 1, \quad f(2\pi) = \sec(2\pi) = 1.
\]
So, \( f(0) = f(2\pi) \) is satisfied.

### Conclusion
Although \( f(0) = f(2\pi) \), Rolle's Theorem is **not applicable** to \( f(x) = \sec x \) on the interval \([0, 2\pi]\) because \( f(x) \) is not continuous on the entire interval \([0, 2\pi]\) and not differentiable on the open interval \( (0, 2\pi) \).

Would you like more details or have any questions about this?

Here are 5 related questions to expand your understanding:
1. What are the conditions required for Rolle's Theorem to hold?
2. How does continuity and differentiability affect the applicability of Rolle's Theorem?
3. Can Rolle's Theorem be applied to other trigonometric functions over specific intervals?
4. What is the geometric interpretation of Rolle's Theorem?
5. How does Rolle's Theorem relate to the Mean Value Theorem?

**Tip:** When checking the applicability of Rolle's Theorem, always start by verifying the continuity of the function over the entire closed interval.

Examine if Rolle’s theorem is applicable for f(x) = sec x in [0, 2π].

Analyze whether Rolle's Theorem can be applied to the function f(x) = sec x in the interval [0, 2π]. This solution covers concepts such as continuity, differentiability, and trigonometric functions, explaining why the theorem does not hold in this case.

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