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MVT-like integral problem involving id.f

Source: Romanian District Olympiad 2009, Grade XII, Problem 3

October 8, 2018
functionintegrationcalculus

Problem Statement

Let f:[0,1]⟶R f:[0,1]\longrightarrow\mathbb{R} be a continuous function such that ∫01(x−1)f(x)dx=0. \int_0^1 (x-1)f(x)dx =0. Show that:
a) There exists a∈(0,1) a\in (0,1) such that ∫0axf(x)dx=0. \int_0^a xf(x)dx =0.
b) There exists b∈(0,1) b\in (0,1) so that ∫0bxf(x)dx=bf(b). \int_0^b xf(x)dx=bf(b).
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