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To determine whether the function satisfies the Mean Value Theorem (MVT) on the interval \([a, b]\), let's analyze the graph and address the following key points:

### Step 1: Check the Hypotheses of MVT
The Mean Value Theorem requires two conditions to be met on the interval \([a, b]\):
1. The function must be **continuous** on the closed interval \([a, b]\).
2. The function must be **differentiable** on the open interval \((a, b)\).

From the graph, if the function is a continuous curve without any breaks or jumps on \([a, b]\) and there are no sharp corners or cusps that indicate non-differentiability, then it would satisfy these hypotheses.

### Step 2: Does the Graph Satisfy the Conclusion of MVT?
If the hypotheses are satisfied, then MVT states that there exists at least one point \(c\) in the open interval \((a, b)\) such that:
\[
f'(c) = \frac{f(b) - f(a)}{b - a}
\]

This means there is at least one point where the slope of the tangent line equals the slope of the secant line connecting \(f(a)\) and \(f(b)\). Based on the graph:
- Identify \(f(a)\) and \(f(b)\), then calculate the slope \(\frac{f(b) - f(a)}{b - a}\).
- Look for where the tangent line’s slope equals this value.

### Analysis of the Graph
From the graph, we observe the following:
- There is a clear line segment between points \(a\) and \(n\), and between \(n\) and \(p\), suggesting the function is piecewise linear. The function is continuous.
- Differentiability might be an issue at points \(n\) and \(p\) if there are sharp corners or changes in direction.

Given the function's piecewise linear nature, it appears to be differentiable except possibly at \(n\) and \(p\). Therefore, the function **likely does not satisfy the differentiability condition** throughout \((a, b)\) if there are sharp points, so MVT might not apply.

### Conclusion:
1. **Hypotheses**: It might fail the differentiability requirement due to the sharp points at \(n\) and \(p\). Thus, MVT may not apply.
2. **Conclusion**: If MVT doesn't apply, we can't guarantee a point \(c\) exists as per MVT. However, if it were differentiable, the slope \(f'(c)\) would match the secant line’s slope.

Would you like a more detailed breakdown, or do you have any specific questions?

### Further Exploration:
1. What are the exact coordinates of \(a\) and \(b\) to compute the secant line's slope?
2. Are the points \(n\) and \(p\) sharp turns, or do they just appear that way?
3. How can we determine if a function is differentiable at a specific point on a graph?
4. What is the geometric interpretation of the Mean Value Theorem?
5. Can a function be continuous but not differentiable, and how does that impact MVT?

### Tip:
When applying the Mean Value Theorem, always double-check for continuity and differentiability across the specified interval. If there’s a sharp corner or discontinuity, MVT may not hold.

Does the function graphed satisfy the hypotheses of the Mean Value Theorem on the interval [a, b]?

Analyze whether the given function satisfies the Mean Value Theorem by checking for continuity and differentiability. Learn how to apply the MVT to find points where the tangent line slope equals the secant line slope on the interval [a, b].

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