Prove-that-Artimetric-mean-Geometric-mean-a-b-2-ab-

Question Number 151448 by mathdanisur last updated on 21/Aug/21

Prove that :

A rtimetric mean ⩾ G eometric mean

\frac{a + b}{2} ⩾ \sqrt{ab}

Commented by puissant last updated on 21/Aug/21

\forall (a, b) \in R^{2},

{(\sqrt{a} - \sqrt{b})}^{2} ⩾ 0

\Rightarrow a - 2 \sqrt{a b} + b ⩾ 0

\Rightarrow 2 \sqrt{a b} ⩽ a + b

\Rightarrow \sqrt{a b} ⩽ \frac{a + b}{2} . .

Commented by mathdanisur last updated on 21/Aug/21

Thank you Ser

Answered by nimnim last updated on 21/Aug/21

This inequality hold when a and b are positive .

Since a and b are positive,

\sqrt{a} = x and \sqrt{b} = y (say)

then {(x - y)}^{2} ⩾ 0

\Rightarrow x^{2} + y^{2} - 2 xy ⩾ 0

\Rightarrow \frac{x^{2} + y^{2}}{2} ⩾ xy

\Rightarrow \frac{a + b}{2} ⩾ \sqrt{ab} ★

Commented by mathdanisur last updated on 21/Aug/21

Thank you Ser