= Solution
Choose a point of $K$ with both coordinates positive, so its product is positive. The set $K\cap[0,\infty)^2$ is nonempty and compact; the <continuous function> $xy$ therefore attains its maximum there. A maximizer cannot lie on either axis, where the product is zero. It consequently belongs to $K\cap P^+$ and also maximizes the product on that set.
For uniqueness, the function $\Phi(x,y)=\log x+\log y$ is <strictly concave> on the positive quadrant: its <Hessian> is $\operatorname{diag}(-1/x^2,-1/y^2)$, <negative definite>. If two distinct positive points attained the same maximal product, their midpoint would belong to $K\cap P^+$ by <convexity> and <strict concavity> would give a larger value of $\Phi$, hence a larger product. This contradiction proves \b[existence and uniqueness of the positive product maximizer]. This argument is <positive-product maximization on a convex set>; <compactness> should be applied to the closed quadrant intersection, not directly to the potentially noncompact set $K\cap P^+$.
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