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Extension: What Is Interpolation?

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Problems 1

Is there a nonzero continuous function f(x)f(x)f(x) such that ∫01f(t)(1−t)nn! dt=0\int_0^1 f(t) \frac{(1-t)^n}{n!}\,dt = 0∫01f(t)n!(1−t)ndt=0 for all positive integers nnn?