Question

If $ax+\frac{b}{x}\ge c,\forall x>0$ and $a,b,c$ are constants then

• A. $ab\ge \frac{c^2}{4}$
• B. $ab\le \frac{c^2}{4}$
• C. $bc\ge \frac{a^2}{4}$
• D. $ac\ge \frac{b^2}{4}$

Answer 1

The correct option is A $ab\ge \frac{c^2}{4}$

Solution:- (A) $ab\ge \frac{c^2}{4}$

$ax+\frac{b}{x}\ge c.....\left(1\right)$

By A.M - G.M. inequality,

$\frac{ax+\frac{b}{x}}{2}\ge \sqrt{ax.\frac{b}{x}}$

$\Rightarrow ax+\frac{b}{x}\ge 2\sqrt{ab}$

$2\sqrt{ab}\ge c\left[\text{From}\left(1\right)\right]$

Squaring both sides, we get

$4ab\ge c^2$

$ab\ge \frac{c^2}{4}$

Hence the correct answer is $ab\ge \frac{c^2}{4}$.