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.