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


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.

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

Explore how the Intermediate Value Theorem applies to the continuous function f(x) on [0, 6], where f(x) = 3 only at x = 1 and x = 5, and f(4) = 2. Analyze which statements about f(2), f(0), and f(6) must be true based on continuity.

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