We are given that $f$ is continuous on the interval $[0, 6]$ and that the only solutions to the equation $f(x) = 3$ occur at $x = 1$ and $x = 5$. This implies that between $x = 1$ and $x = 5$, the function $f(x)$ does not equal 3 anywhere else. Additionally, we know that $f(4) = 5$, which is greater than 3.

We will now analyze each of the three statements:

Statement (i): $f(2) > 3$

• The function is continuous, and the only points where $f(x) = 3$ are $x = 1$ and $x = 5$.
• From the Intermediate Value Theorem (IVT), since $f(1) = 3$ and $f(4) = 5$, $f(x)$ must be greater than 3 between $x = 1$ and $x = 4$. Hence, $f(2) > 3$.

So, statement (i) is true.

Statement (ii): $f(0) < 3$

• Since $f(x) = 3$ only at $x = 1$, and $f(x)$ must be either greater than or less than 3 outside the point $x = 1$.
• If $f(0) \geq 3$, the continuity of $f$ would force the function to reach 3 before $x = 1$, violating the condition that $f(x) = 3$ only at $x = 1$.
• Therefore, $f(0)$ must be less than 3 to avoid crossing 3 anywhere between $x = 0$ and $x = 1$.

So, statement (ii) is true.

Statement (iii): $f(6) < 3$

• Similarly, $f(x) = 3$ only at $x = 5$, and for $f(x)$ to stay above or at 3 after $x = 5$, it would have to cross 3 again after $x = 5$, which would contradict the given condition.
• Therefore, $f(6)$ must be less than 3 to ensure that the function does not cross the value 3 after $x = 5$.

So, statement (iii) is true.

Conclusion:

All three statements must be true. Therefore, the correct answer is that statements (i), (ii), and (iii) are all true.

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Do you want more details or have any questions? Here are some additional questions to explore:

1. Can the Intermediate Value Theorem be applied to any function that is not continuous?
2. What happens if $f(2) = 3$? Would this affect the results?
3. How would the situation change if $f(x) = 3$ had more solutions on $[0, 6]$?
4. Could there be another point where $f(x) = 5$ apart from $x = 4$?
5. What would happen if $f(x)$ was discontinuous at any point in the interval?

Tip: The Intermediate Value Theorem is a powerful tool when analyzing continuous functions, as it guarantees that a function takes on all values between any two points on its graph.