We are given that $f$ is continuous on the interval $[0, 6]$, and the only solutions to the equation $f(x) = 3$ are $x = 1$ and $x = 5$. This means that the function only takes the value 3 at these two points, and nowhere else on the interval.

Additionally, we are told that $f(4) = 2$, which provides additional information about the behavior of the function.

Let's analyze each statement:

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(i) $f(2) < 3$

• Since $f(x) = 3$ only at $x = 1$ and $x = 5$, and the function is continuous, it must transition through values lower than 3 when moving from $f(1) = 3$ to $f(4) = 2$.
• Specifically, between $x = 1$ and $x = 4$, $f(x)$ must decrease because at $x = 1$, $f(1) = 3$, and at $x = 4$, $f(4) = 2$.
• Therefore, $f(2)$ must be less than 3.

Thus, statement (i) is true.

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(ii) $f(0) > 3$

• Since $f(1) = 3$ and there are no other points where $f(x) = 3$ besides $x = 1$ and $x = 5$, we need to check the behavior of the function at $x = 0$.
• The continuity of $f$ suggests that moving from $x = 0$ to $x = 1$, the function must either stay above 3 or decrease to 3 as $x$ approaches 1.
• However, there is no explicit information about $f(0)$, so we cannot conclude that $f(0)$ is greater than 3 without further details.

Thus, statement (ii) is not necessarily true.

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(iii) $f(6) < 3$

• Similarly, since $f(5) = 3$ and there are no other solutions to $f(x) = 3$ on the interval, we know the function moves away from 3 for $x > 5$.
• Given that $f$ is continuous and that $f(x) = 3$ only at $x = 5$, the function must take values less than 3 beyond $x = 5$, specifically at $x = 6$.

Thus, statement (iii) is true.

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Conclusion:

The statements that must be true are:

• (i) $f(2) < 3$
• (iii) $f(6) < 3$

Statement (ii) cannot be determined based on the given information.

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Do you want more details or have any questions?

Here are 5 related questions to consider:

1. How does the Intermediate Value Theorem apply in this situation?
2. What would happen if $f(4)$ were greater than 3 instead of 2?
3. Can we deduce the exact value of $f(0)$ with additional assumptions?
4. How does the continuity of $f$ affect its values between $x = 1$ and $x = 5$?
5. What would the function's graph likely look like based on the given information?

Tip: Continuity plays a crucial role in determining behavior between known values of a function, especially when applying the Intermediate Value Theorem.