Show further that the Cauchy distribution with p(x) = 1 1 π 1 + x² ?? is a solution for the case f(x) = ln(1 + x²) and c = 2ln 2, and obtain the Lagrange multipliers in this case. You may assume that the methods of the calculus of variations apply on the infinite interval (–∞, ∞). The values of the following integrals may be useful: ୮ x dx 1 1+x² = π, مر In(1 + x²) dx = 2π In 2. 1+x² For a probability density function p(x) > 0 on the interval (-∞, ∞) the entropy functional S[p] is given by S[p] = − (** dx p(x) logp(x). - L Suppose that the function p(x) is subject to the constraints Lo dr p(x) dx p(x) = 1 and L dx f(x)p(x) = c, -00 -∞ where f(x) is a fixed function and c is a constant. stationary path for S[p] is given by - p(x) = exp(−1 — λ — µƒ(x)), where and μ are Lagrange multipliers.