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The problem asks to determine the values of \(a\), \(b\), and \(c\) such that the function \(f(x)\) satisfies the hypotheses of the Mean Value Theorem (MVT) on the interval \([0, 3]\).

The function \(f(x)\) is defined piecewise:

\[
f(x) =
\begin{cases}
3 & \text{if } x = 0 \\
ax + b & \text{if } 0 < x \leq 1 \\
x^2 + 2x + c & \text{if } 1 < x \leq 3
\end{cases}
\]

To satisfy the Mean Value Theorem, the function must be:
1. Continuous on the closed interval \([0, 3]\).
2. Differentiable on the open interval \((0, 3)\).

Let's go step-by-step to impose the conditions for continuity and differentiability.

### Step 1: Continuity at \(x = 1\)
For continuity at \(x = 1\), the two pieces on either side of \(x = 1\) must be equal at \(x = 1\):
\[
ax + b = x^2 + 2x + c \quad \text{at} \quad x = 1
\]
Substitute \(x = 1\) into both expressions:
\[
a(1) + b = 1^2 + 2(1) + c
\]
\[
a + b = 1 + 2 + c
\]
\[
a + b = 3 + c \quad \text{(Equation 1)}
\]

### Step 2: Continuity at \(x = 0\)
For continuity at \(x = 0\), the first piece must match the second piece at \(x = 0\):
\[
f(0) = 3
\]
Substitute \(x = 0\) into the second piece:
\[
a(0) + b = 3
\]
\[
b = 3 \quad \text{(Equation 2)}
\]

### Step 3: Differentiability at \(x = 1\)
For differentiability at \(x = 1\), the derivatives of the two pieces must match at \(x = 1\).

Derivative of \(ax + b\) is:
\[
f'(x) = a
\]
Derivative of \(x^2 + 2x + c\) is:
\[
f'(x) = 2x + 2
\]
At \(x = 1\), the derivatives must be equal:
\[
a = 2(1) + 2
\]
\[
a = 4 \quad \text{(Equation 3)}
\]

### Step 4: Solve the System of Equations
We now have three equations:
1. \(a + b = 3 + c\)
2. \(b = 3\)
3. \(a = 4\)

Substitute \(a = 4\) and \(b = 3\) into Equation 1:
\[
4 + 3 = 3 + c
\]
\[
7 = 3 + c
\]
\[
c = 4
\]

### Final Answer
Thus, the values are:
\[
a = 4, \quad b = 3, \quad c = 4
\]

Would you like further clarification or have any questions?

### Related Questions
1. What does the Mean Value Theorem state, and why is differentiability required?
2. Can a function be continuous but not differentiable? Provide an example.
3. How do you compute the derivative of a piecewise function?
4. What conditions need to be checked for the Mean Value Theorem besides differentiability?
5. How can you apply the Mean Value Theorem in practical scenarios?

**Tip:** Continuity and differentiability are key properties when working with piecewise functions. Always check transitions at boundary points between different pieces.

Determine the values of a, b, and c such that the function f satisfies the hypotheses of the Mean Value Theorem on the interval [0, 3].

Learn how to determine the values of a, b, and c for a piecewise function such that it satisfies the Mean Value Theorem on the interval [0, 3]. This problem covers key concepts like continuity, differentiability, and the Mean Value Theorem. Suitable for students in Grades 11-12.

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