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A mean value problem \int_0^1 f(x)dx=0 and \int_{0}^{c}xf(x)

Source: Romanian National Olympiad, 2006

April 17, 2006
integrationfunctioncalculusderivativetrigonometryreal analysisreal analysis unsolved

Problem Statement

Let f:[0,1]→Rf: [0,1]\to\mathbb{R} be a continuous function such that ∫01f(x)dx=0. \int_{0}^{1}f(x)dx=0. Prove that there is c∈(0,1)c\in (0,1) such that ∫0cxf(x)dx=0. \int_{0}^{c}xf(x)dx=0. Cezar Lupu, Tudorel Lupu
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