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4

Part of 2019 ISI Entrance Examination

Problems(1)

ISI 2019 : Problem #4

Source: I.S.I. 2019

5/5/2019
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a twice differentiable function such that 12y∫x−yx+yf(t) dt=f(x)∀ x∈R & y>0\frac{1}{2y}\int_{x-y}^{x+y}f(t)\, dt=f(x)\qquad\forall~x\in\mathbb{R}~\&~y>0 Show that there exist a,b∈Ra,b\in\mathbb{R} such that f(x)=ax+bf(x)=ax+b for all x∈Rx\in\mathbb{R}.
isiIndian Statistical Institute2019calculus