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Let f(x) = sin(pi*x) / x² for x > 0. Let x1 < x2 < x3 <... be all the points of local maximum of f, and y1 < y2 < y3 <... be all the points of local minimum of f. Which of the following options is/are correct?

1. |xₙ - yₙ| > 1 for every positive integer n.
2. x₁ < y₁.
3. xₙ belongs to the interval (2n, 2n + 1/2) for every positive integer n.
4. x_(n+1) - xₙ > 2 for every positive integer n.

Correct answer: x_(n+1) - xₙ > 2 for every positive integer n.

Solution

Setting f'(x) = 0 gives tan(pi*x) = pi*x/2. Local maxima of f occur where sin(pi*x) > 0 (intervals (2k-1, 2k) shifted), and local minima where sin(pi*x) < 0. The critical points between consecutive integers are slightly shifted from half-integer values. Analysis shows x_(n+1) - xₙ > 2 for all n.

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