AM–GM inequality

Uncommon card · No. 159232

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 (in which case they are both that number). The simplest non-trivial case is for two non-negative numbers x and y, that is, x + y 2 ≥ x y {\displaystyle {\frac {x+y}{2}}\geq {\sqrt {xy}}} with equality if and only if x = y. This follows from the fact that the square of a real number is always non-negative (greater than or equal to zero) and from the identity (a ± b)2 = a2 ± 2ab + b2:...

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