Let's analyze the given problem:

The function $f$ is continuous on the interval $[0, 6]$ and the only solutions to the equation $f(x) = 3$ are at $x = 1$ and $x = 5$. We also know that $f(4) = 2$. We are asked to determine which of the following statements must be true:

(i) $f(2) > 3$
(ii) $f(0) > 3$
(iii) $f(6) > 3$

Key points to consider:

1. Continuity of $f$: Since $f$ is continuous on $[0, 6]$, we can apply the Intermediate Value Theorem (IVT).
2. Solutions of $f(x) = 3$: The only places where $f(x) = 3$ are at $x = 1$ and $x = 5$.
3. Value of $f(4) = 2$: This tells us that $f(x)$ is below 3 at $x = 4$.

Now, let's reason through the options:

Interval Analysis:

• On the interval $[0, 1]$, we do not have any information about specific values, but we know that $f(x) \neq 3$ except at $x = 1$. By the IVT, $f(x)$ must change from something above 3 or below 3 as it approaches $x = 1$.
• On the interval $[1, 5]$, since $f(4) = 2$ (which is less than 3), $f(x)$ must dip below 3 at some point. At $x = 5$, it returns to 3.
• On the interval $[5, 6]$, there is no specific information provided other than that $f(x) \neq 3$ in this region.

Evaluating the Statements:

(i) $f(2) > 3$:
Between $x = 1$ and $x = 5$, we know that $f(4) = 2$, so at some point the function must drop below 3. Thus, $f(2) > 3$ cannot be guaranteed. This statement is not necessarily true.

(ii) $f(0) > 3$:
Since $f(x) = 3$ only at $x = 1$ and $f$ is continuous, $f(0)$ must either be greater than or less than 3. But there's no information suggesting that it must be greater. Therefore, this is not necessarily true.

(iii) $f(6) > 3$:
On the interval $[5, 6]$, $f(x) \neq 3$, but there is no specific information indicating whether $f(6)$ must be greater than 3. This is also not necessarily true.

Conclusion:

Based on the given conditions, none of the statements must be true, so the correct answer is (C) none of them.

Do you need any further clarifications or more details?

Here are 5 related questions to consider:

1. How does the Intermediate Value Theorem apply in this scenario?
2. What happens to the function $f(x)$ between $x = 1$ and $x = 5$?
3. Could $f(0)$ or $f(6)$ be less than 3, and why?
4. How do we know that $f(2)$ is less than 3 in this case?
5. Can we infer the behavior of $f(x)$ on intervals where $f(x) \neq 3$?

Tip: Always pay close attention to the points where a function equals a certain value, especially in the context of continuity and the Intermediate Value Theorem.