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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**.

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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.

Suppose that f is continuous on [0, 6] and that the only solutions of the equation f(x) = 3 are x = 1 and x = 5. If f(4) = 2, then which of the following statements must be true?
(i) f(2) > 3
(ii) f(0) > 3
(iii) f(6) > 3

This problem involves analyzing a continuous function f on the interval [0,6], using the Intermediate Value Theorem to determine the truth of various inequalities. Suitable for students in Grades 10-12.

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