The minimum value of $ax + by$, where $xy = c^2 $ and a, b, c are positive, is :

$2c\sqrt{ab}$

The correct answer is Option (3) → $2c\sqrt{ab}$

The constraint, $xy=c^2$, which can be written as:

$y=\frac{c^2}{x}$

Substitute $y=\frac{c^2}{x}$ into the objective function

$ax+by=ax+b.\frac{c^2}{x}$

$f(x)=ax+\frac{bc^2}{x}$

$⇒\frac{d}{dx}\left(ax+\frac{bc^2}{x}\right)=a-\frac{bc^2}{x^2}$

$⇒a-\frac{bc^2}{x^2}=0$

$⇒\frac{bc^2}{x^2}=a$

$⇒x=\frac{c\sqrt{b}}{\sqrt{a}}$

and, $y=\frac{c^2}{x}=\frac{c\sqrt{a}}{\sqrt{b}}$

∴ Substitute the values,

$ax+by=c\sqrt{ab}+c\sqrt{ab}$

$=2c\sqrt{ab}$