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Apply to a modified form of <integral f = integral id.f> some sort of mvt

Source: Romanian District Olympiad 2008, Grade XII, Problem 1

October 7, 2018
functionreal analysisFTCMVTIntegralcalculusintegration

Problem Statement

Let f:[0,1]⟶R f:[0,1]\longrightarrow\mathbb{R} be a countinuous function such that ∫01f(x)dx=∫01xf(x)dx. \int_0^1 f(x)dx=\int_0^1 xf(x)dx. Show that there is a c∈(0,1) c\in (0,1) such that f(c)=∫0cf(x)dx. f(c)=\int_0^c f(x)dx.
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