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integral inequality with integral condition, [0,1]->R

Source: SEEMOUS 2008 P4

June 17, 2021
calculusintegrationinequalities

Problem Statement

Let nn be a positive integer and f:[0,1]→Rf:[0,1]\to\mathbb R be a continuous function such that ∫01xkf(x)dx=1\int^1_0x^kf(x)dx=1for every k∈{0,1,…,n−1}k\in\{0,1,\ldots,n-1\}. Prove that ∫01f(x)2dx≥n2.\int^1_0f(x)^2dx\ge n^2.
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