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To solve sub-problem 3, we need to determine the real parameters \( a \) and \( b \) such that the function \( f \) satisfies the conditions for applying Lagrange’s Mean Value Theorem (MVT) on the interval \( [-1, 1] \). 

### Problem Statement for Sub-part 3

The function \( f : [-1,1] \to \mathbb{R} \) is defined as:
\[
f(x) = \begin{cases} 
x^4 + ax + 2, & x \in [-1,0] \\
b + \ln(1 + x^2), & x \in [0,1]
\end{cases}
\]

To apply Lagrange’s Mean Value Theorem on \( [-1, 1] \), \( f \) must be:
1. **Continuous on the closed interval** \( [-1, 1] \),
2. **Differentiable on the open interval** \( (-1, 1) \).

The main task is to ensure continuity of \( f(x) \) at \( x = 0 \), since this is the boundary between the two pieces of \( f \). 

### Step 1: Ensuring Continuity at \( x = 0 \)
For \( f(x) \) to be continuous at \( x = 0 \):
\[
\lim_{x \to 0^-} f(x) = f(0) = \lim_{x \to 0^+} f(x).
\]

1. **Left-hand limit at \( x = 0 \):**
   \[
   \lim_{x \to 0^-} f(x) = f(0^-) = 0^4 + a \cdot 0 + 2 = 2.
   \]

2. **Right-hand limit at \( x = 0 \):**
   \[
   \lim_{x \to 0^+} f(x) = f(0^+) = b + \ln(1 + 0^2) = b + \ln(1) = b.
   \]

For continuity at \( x = 0 \), we require:
\[
2 = b.
\]

### Step 2: Conclusion
To satisfy the conditions for Lagrange's Mean Value Theorem on \( [-1, 1] \), we need \( b = 2 \). The value of \( a \) is not restricted by the continuity requirement, so it can be any real number.

### Final Answer
- \( b = 2 \)
- \( a \in \mathbb{R} \) (any real value)

Would you like more details on the differentiability conditions or any additional clarifications?

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### Further Questions
1. How do we verify differentiability of piecewise functions like this?
2. What is the statement of Lagrange’s Mean Value Theorem?
3. How does continuity impact the application of MVT?
4. Can \( a \) affect the differentiability of \( f \) on \( (-1, 1) \)?
5. How would we find the mean value of \( f \) if MVT applies?

#### Tip
For piecewise functions, always check continuity at the boundary points to ensure smooth transitions across intervals.

Find the values of parameters a and b for a piecewise function to apply Lagrange's Mean Value Theorem on [-1,1]. This problem explores continuity and differentiability conditions, particularly at x=0 where the function's definition changes.

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