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A function f(x) is continuous in the interval [0,2]. It is known that f(0) = f(2) = -1 and f(1) = 1. Which one of the following statements must be true?

1. There exists a y in the interval (0,1) such that f(y) = f(y + 1)
2. For every y in the interval (0,1), f(y) = f(2 - y)
3. The maximum value of the function in the interval (0,2) is 1
4. There exists a y in the interval (0,1) such that f(y) = -f(2 - y)

Correct answer: There exists a y in the interval (0,1) such that f(y) = -f(2 - y)

Solution

This statement is true due to the Intermediate Value Theorem and the properties of continuous functions. Since f(0) = -1 and f(2) = -1, and f(1) = 1, there must be some point in (0,1) where the function takes on the value that is the negative of its value at the corresponding point in (1,2), ensuring that f(y) equals -f(2 - y) for some y in that interval.

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