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 $f(2)$?

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 $\boxed{13}$.

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.