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Integral problem using periodicity

Source: RomNO 2003, grade xii, p.3

August 27, 2019
functioncalculusintegration

Problem Statement

Let be a continuous function f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} that has the property that xf(x)≥∫0xf(t)dt, xf(x)\ge \int_0^x f(t)dt , for all real numbers x. x. Prove that
a) the mapping x↦1x∫0xf(t)dt x\mapsto \frac{1}{x}\int_0^x f(t) dt is nondecreasing on the restrictions R<0 \mathbb{R}_{<0 } and R>0. \mathbb{R}_{>0 } .
b) if ∫xx+1f(t)dt=∫x−1xf(t)dt, \int_x^{x+1} f(t)dt=\int_{x-1}^x f(t)dt , for any real number x, x, then f f is constant.
Mihai Piticari
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