= Solution
For a point in the first quadrant, $xy$ is the <area> of the axis-aligned rectangle with opposite vertices at the origin and $(x,y)$. Thus the maximum says that the largest such quarter-rectangle with its outer corner on the <ellipse> has <area> one.
Equivalently, reflect this rectangle in both axes. Its four vertices $(\pm|x|,\pm|y|)$ lie on the <ellipse>, and its full <area> is $4|xy|$. Therefore
$$
\boxed{\text{the maximum area of an axis-aligned rectangle inscribed in the ellipse is }4}.
$$
Its width is $1/\sqrt2$ and its height is $4\sqrt2$.
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