Let f(x)=|x|. Which of the following statements is correct?




Question 1 Select one:

The function f satisfies the hypotheses of Rolle's theorem on the interval [−1,1].
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The function f does not satisfy the hypotheses of Rolle's theorem on the interval [−1,1], since f is not continuous at x=0.
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The function f does not satisfy the hypotheses of Rolle's theorem on the interval [−1,1], since f is not differentiable at x=0.
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The function f does not satisfy the hypotheses of Rolle's theorem on the interval [−1,1], since f(−1)≠f(1).


Let's analyze each option based on Rolle's Theorem. For a function \( f \) to satisfy Rolle's theorem on a closed interval \([a, b]\), it must meet three conditions:

1. **Continuity** on \([a, b]\).
2. **Differentiability** on the open interval \((a, b)\).
3. **Equal values at the endpoints**: \( f(a) = f(b) \).

Given the function \( f(x) = |x| \) on the interval \([-1, 1]\):

1. \( f(x) \) **is continuous** on \([-1, 1]\), since absolute value functions are continuous everywhere.
2. \( f(x) \) **is not differentiable** at \( x = 0 \), as there is a cusp at \( x = 0 \).
3. \( f(-1) = |{-1}| = 1\) and \( f(1) = |1| = 1\), so \( f(-1) = f(1) \).

Since the function fails the differentiability condition (it’s not differentiable at \( x = 0 \)), **the correct answer is**:

> **The function \( f \) does not satisfy the hypotheses of Rolle's theorem on the interval \([-1,1]\), since \( f \) is not differentiable at \( x = 0 \).**

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### Related Questions
1. How do we determine if a function is differentiable at a point?
2. What are other examples of functions that are continuous but not differentiable?
3. How would the result change if the interval was \([0,1]\) instead of \([-1,1]\)?
4. Why is continuity alone not enough to apply Rolle's Theorem?
5. Can Rolle's Theorem be applied to functions with endpoints not included in the interval?

### Tip
When checking if Rolle's Theorem applies, always verify differentiability over the entire open interval—not just at the endpoints.

Let f(x)=|x|. Which of the following statements is correct?

1. The function f satisfies the hypotheses of Rolle's theorem on the interval [−1,1].
2. The function f does not satisfy the hypotheses of Rolle's theorem on the interval [−1,1], since f is not continuous at x=0.
3. The function f does not satisfy the hypotheses of Rolle's theorem on the interval [−1,1], since f is not differentiable at x=0.
4. The function f does not satisfy the hypotheses of Rolle's theorem on the interval [−1,1], since f(−1)≠f(1).

Explore why the function f(x) = |x| does not satisfy the conditions of Rolle's theorem on the interval [-1,1]. This problem covers key calculus concepts including continuity, differentiability, and Rolle's theorem, making it ideal for Grade 12 or college students.

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