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Given two continuous functions f and g defined on the interval [0, 1] such that the maximum value of f(x) over [0, 1] equals the maximum value of g(x) over [0, 1], which of the following statements is true?

1. There exists a point c in [0, 1] where (f(c))² + 3f(c) equals (g(c))² + 3g(c).
2. There exists a point c in [0, 1] where (f(c))² + f(c) equals (g(c))² + 3g(c).
3. There exists a point c in [0, 1] where (f(c))² + 3f(c) equals (g(c))² + g(c).
4. There exists a point c in [0, 1] where (f(c))² equals (g(c))².

Correct answer: There exists a point c in [0, 1] where (f(c))² + 3f(c) equals (g(c))² + 3g(c).

Solution

The correct answer is that there exists a point c in [0, 1] where (f(c))² + 3f(c) equals (g(c))² + 3g(c) because both functions reach the same maximum value at some point in the interval, resulting in f(c) = g(c), which satisfies the given equation.

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