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Suppose f: R → (0, 1) is a continuous function. Which of the following functions equals zero at least at one point within the interval (0, 1)?

1. f(x) + ∫₀ˣ f(t) sin(t) dt
2. eˣ − ∫₀ˣ f(t) sin(t) dt
3. x² − f(x)
4. x − ∫₀ˣ f(t) cos(t) dt

Correct answer: x² − f(x)

Solution

The function x² - f(x) must equal zero at some point in the interval (0, 1) due to the intermediate value theorem, which states that a continuous function must take on all values between its maximum and minimum.

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