Question

If $A$ and $G$ be A.M. and G.M. of two given positive real numbers $a$ and $b$ respectively, then $A$ and $G$ are related as

• A. $A \geq G$
• B. $G \geq A$
• C. $A=G$
• D. $A=-G$

Answer 1

Answer: (A)

Let $A$ and $G$ be A.M and G.M. of two given positive real numbers $a$ and $b$ respectively.
Then, $ A=\frac{a+b}{2}$ and $G=\sqrt{a b}$
Thus, we have
$A-G =\frac{a+b}{2}-\sqrt{a b}=\frac{a+b-2 \sqrt{a b}}{2}$
$ =\frac{(\sqrt{a}-\sqrt{b})^2}{2} \geq 0$
Hence, $ A-G \geq 0 \Rightarrow A \geq G$