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The problem in the image reads:

**Question 1 (a):**

Suppose \( f \) is continuous on \([-2, 2]\) and differentiable on \((-2, 2)\) such that \( f(-2) = 5 \) and \( |f'(x)| \leq 2 \) for all \( x \in (-2, 2) \). What will be the largest possible value of \( f(2) \)?

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### Solution Approach:

The problem uses information about the derivative \( f'(x) \), and asks for the largest possible value of \( f(2) \). This can be solved using **differential inequalities** and **estimating the change** in the function value based on the bound on \( |f'(x)| \).

1. We are given that \( |f'(x)| \leq 2 \) for all \( x \) in \( (-2, 2) \). This means that the slope of the function (the rate of change) is bounded by 2. In other words, the function can increase or decrease by at most 2 units for each unit of change in \( x \).

2. The function starts at \( f(-2) = 5 \). We need to find the largest possible value of \( f(2) \).

3. Since the maximum rate of change of \( f(x) \) is \( 2 \), the most \( f(x) \) can increase as \( x \) moves from \( -2 \) to \( 2 \) is \( 4 \times 2 = 8 \) units (as the distance between \( -2 \) and \( 2 \) is 4).

4. Therefore, the largest possible value of \( f(2) \) is:
   \[
   f(2) = f(-2) + 8 = 5 + 8 = 13
   \]

Thus, the largest possible value of \( f(2) \) is \( \boxed{13} \).

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Do you need more details or have any other questions?

Here are 5 related questions to deepen your understanding:
1. What would the smallest possible value of \( f(2) \) be, given the same conditions?
2. How would the solution change if the bound on \( |f'(x)| \) was reduced to 1 instead of 2?
3. Why is the continuity and differentiability of \( f \) on the given intervals important in this problem?
4. Can you generalize this method to estimate the maximum value of a function if the bound on the derivative changes over the interval?
5. How does this problem relate to the Mean Value Theorem?

**Tip:** Always look at the derivative constraints when estimating function values; they provide a clue about how fast or slow the function can change.

Suppose f is continuous on [-2, 2] and differentiable on (-2, 2) such that f(-2) = 5 and |f'(x)| ≤ 2 for all x in (-2, 2). What will be the largest possible value of f(2)?

This problem explores how to estimate the maximum possible value of f(2) given that f is continuous and differentiable on [-2, 2] with a bound on |f'(x)|. Suitable for undergraduate calculus students, the problem applies the Mean Value Theorem and concepts of differential inequalities.

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