Among all rectangles of a given area the square has the least perimeter.
Among all rectangles of a given perimeter the square has the largest area.
This duality of the formulation carries over to the AM-AG inequality.
For positive a, b that satisfy a + b = 2, ab ≤ 1.
For positive a, b that satisfy ab = 1, a + b ≥ 2.
While trivial, it is often useful while solving problems to keep this interpretation in mind. Here's one example from the 1935 Moscow Mathematical Olympiad:
Find all real solutions of the following system:
x + y = 2
xy - z² = 1.
Since x + y = 2, xy ≤ 1 so that
1 = xy - z² ≤ 1 - z² < 1,
unless z = 0. To avoid a contradiction (1 < 1), we have to accept