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To analyze this problem with respect to the Mean Value Theorem (MVT), let's go through each component carefully:

### 1. Checking the Hypotheses of the Mean Value Theorem (MVT)
The Mean Value Theorem requires that:
- The function \( f \) is **continuous** on the closed interval \([a, b]\).
- The function \( f \) is **differentiable** on the open interval \((a, b)\).

From the graph:
- The function \( f(x) \) appears to be continuous on \([a, b]\).
- There are points where the slope of \( f(x) \) changes abruptly, indicating **corners** in the graph. Specifically, the function is **not differentiable** at points where the slope changes abruptly, which likely includes points such as \( n \) and \( p \).

Since \( f(x) \) is not differentiable at all points within \((a, b)\), the MVT **does not apply**.

### Conclusion:
- **Does the function satisfy the hypotheses of the Mean Value Theorem on \([a, b]\)?** No, because it is not differentiable at all points on \((a, b)\).
- **Does it satisfy the conclusion?** Since the hypotheses are not satisfied, the conclusion of the MVT does not apply here either.

#### Answer Summary:
1. **Hypotheses**: No.
2. **Conclusion**: No.
3. **Point \( c \)**: Not applicable, as the MVT does not hold for this function.

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Would you like more details on this solution or have any other questions?

### Related Questions
1. What are the conditions required for differentiability of a function on an interval?
2. Can the Mean Value Theorem be applied to a piecewise linear function?
3. How do we determine if a function has corners or discontinuities?
4. What other theorems might apply when the Mean Value Theorem does not?
5. How can we interpret the conclusion of the MVT in terms of average rate of change?

#### Tip:
When checking if a function meets the Mean Value Theorem, always verify both continuity and differentiability across the entire interval.

Does this function satisfy the hypotheses of the Mean Value Theorem on the interval [a, b]? If so, at what point c is f'(c) = (f(b) - f(a)) / (b - a)?

Analyze whether a piecewise linear function satisfies the hypotheses of the Mean Value Theorem on the interval [a, b]. Learn how continuity and differentiability play roles in verifying MVT applicability and understand how to determine the average rate of change on the interval.

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