Inequality of arithmetic and geometric means

In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same.

The simplest non-trivial case — i.e., with more than one variable — for two non-negative numbers $x$ and $y$ , is the statement that

with equality if and only if $x = y$ . This case can be seen from the fact that the square of a real number is always non-negative (greater than or equal to zero) and from the elementary case $(a \pm b) 2 = a 2 \pm 2 ab + b 2$ of the binomial formula:

In other words $(x + y) 2 \geq 4 xy$ , with equality precisely when $(x - y) 2 = 0$ , i.e. $x = y$ .

For a geometrical interpretation, consider a rectangle with sides of length $x$ and $y$ , hence it has perimeter $2 x + 2 y$ and area $xy$ . Similarly, a square with all sides of length $\sqrt xy$ has the perimeter $4 \sqrt xy$ and the same area as the rectangle. The simplest non-trivial case of the AM–GM inequality implies for the perimeters that $2 x + 2 y \geq 4 \sqrt xy$ and that only the square has the smallest perimeter amongst all rectangles of equal area.

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