For a probability density function p(x) > 0 on the interval (00, 00) the entropy functional S|p| is given by Sip] = / "~ dz p(z) log p(2). Suppose that the function p(x) is subject to the constraints f dzx p(z) =1 and / dz f(z)p(z) = c, where f(z) is a fixed function and c is a constant. stationary path for S[p| is given by p(z) = exp(1 A puf(z)), where A and p are Lagrange multipliers. Show further that the Cauchy distribution with is a solution for the case f(z) = In(1 + z%) and = 2In2, and obtain the Lagrange multipliers in this case. You may assume that the methods of the calculus of variations apply on the infinite interval (oo, 00). The values of the following integrals may be useful: 0 In(1 +z?%) / dx 1+$2 f dx 172 =2rln2