MathDB

ISI 2019 : Problem #4

Source: I.S.I. 2019

May 5, 2019

isiIndian Statistical Institute2019calculus

Problem Statement

Let

f:\mathbb{R}\to\mathbb{R}

be a twice differentiable function such that

\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\in\mathbb{R}

such that

f(x)=ax+b

for all

x\in\mathbb{R}