MathDB
Integral inequlity

Source: Romanian national olympiad

February 20, 2006
calculusintegrationLaTeXfunctionreal analysisreal analysis unsolved

Problem Statement

Prove that: ∫−11f2(x)dx≥12(∫−11f(x)dx)2+32(∫−11xf(x)dx)2\int_{-1}^1f^2(x)dx\ge \frac 1 2 (\int_{-1}^1f(x)dx)^2 +\frac 3 2(\int_{-1}^1xf(x)dx)^2 Please give a proof without using even and odd functions. (the oficial proof uses those and seems to be un-natural) :D
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