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Concave function

Strictly concave function

On the positive orthant,
$$
F(x)=\sum_{j=1}^n\log x_j
$$
is strictly concave. Its unique maximizer $x^*$ on a compact convex feasible set satisfies the first-order inequality
$$
\nabla F(x^*)\cdot(x-x^*)\leq0,
$$
equivalently
$$
\sum_{j=1}^n\frac{x_j}{x_j^*}\leq n.
$$


Product maximizer on a compact convex subset of the positive orthant

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 2H / Solution

Product maximizer on a compact convex subset of the positive orthant

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Product maximizer on a compact convex subset of the positive orthant

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