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The sign of integration f(x)x^k cannot be alternating

Source: 2021 CIIM, P6

November 2, 2021
functionreal analysiscalculusintegration

Problem Statement

Let 0≤a<b0 \le a < b be real numbers. Prove that there is no continuous function f:[a,b]→Rf : [a, b] \to \mathbb{R} such that \int_a^b f(x)x^{2n} \mathrm dx>0   \text{and}   \int_a^b f(x)x^{2n+1} \mathrm dx <0 for every integer n≥0n \ge 0.
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