Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 19 / b / Solution

Use $\mathcal{L}=-{\sum }_i{x}_i\log x_i-\lambda ({\sum }_i{x}_i-1)$. Differentiating with respect to each positive $x_i$ givesEvery $x_i$ is therefore equal; normalization forcesThe Hessian matrix of $f$ is diagonal with entries $-1/x_i$, so its quadratic form is strictly negative for every nonzero perturbation. In particular it is negative along the tangent space ${\sum }_i\delta x_i=0$, making this the unique global maximum on the positive probability simplex. This is the maximum entropy distribution on a finite set, using natural logarithms.