Question

If a and b are two different positive real numbers, then which of the following relation is true.

• A. $2\sqrt{ab}>a+b$
• B. $2\sqrt{ab}<a+b$
• C. $2\sqrt{ab}=a+b$
• D. none of these

Answer 1

The correct option is B

$2\sqrt{ab}<a+b$

Finding the which of the following is a true relation:

We know $AM\ge GM$

$\begin{array}{rcl}\frac{(a+b)}{2}& \ge & \sqrt{\left(ab\right)}\\ (a+b)& \ge & 2\sqrt{\left(ab\right)}\\ \Rightarrow 2\sqrt{(ab})& <& a+b\end{array}$

Hence option (B) is the answer.