= Solution
Use $\mathcal L=-\sum_i x_i\log x_i-\lambda(\sum_i x_i-1)$. Differentiating with respect to each positive $x_i$ gives
$$
-\log x_i-1-\lambda=0.
$$
Every $x_i$ is therefore equal; normalization forces
$$
\boxed{x_i=\frac1n\quad\text{for all }i},\qquad
\boxed{f_{\rm stationary}=\log n}.
$$
The <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.
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