Question

If ${\log }_{10}x+{\log }_{10}y\ge 2$, then the smallest possible value of $x+y$ will be

• A. $10$
• B. $30$
• C. $20$
• D. none of these

Answer 1

The correct option is C $20$

${\log }_{10}x+{\log }_{10}y\ge 2$
$\Rightarrow xy\ge 100$ ...(1)
If $x>0$ & $y>0$
Since $A.M.\ge G.M$
$\Rightarrow \frac{x+y}{2}\ge \sqrt{xy}$
$\Rightarrow \frac{x+y}{2}\ge \sqrt{100}$ ....{ From 1}
$\Rightarrow x+y\ge 20$
Ans: C