Question

Let $x$ and $y$ be two real variables such that $x>0$ and $xy=1.$ Find the minimum value of $x+y$

Answer 1

Let $z=x+y=x+\frac{1}{x}$ (as $xy=1)$
$\frac{dz}{dx}=1-\frac{1}{x^2}$
For minima $\frac{dz}{dx}=0\Rightarrow 1-\frac{1}{x^2}=0$
$\Rightarrow x=\pm 1$ but given $x>0\therefore x=1$
$\Rightarrow z_{min}=1+1=2$