Question

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

• A. 2
• B. 3
• C. 4
• D. ) 0

Answer 1

Answer: (A)

2

Answer 2

$xy = 1$ ⇒ $y = \frac{1}{x}$ and let $z = x + y$

$z = x + \frac{1}{x}$ ⇒ $\frac{dz}{dx} = 1 - \frac{1}{x^{2}}$ ⇒ $\frac{dz}{dx} = 0$ ⇒ $1 - \frac{1}{x^{2}} = 0$

⇒ $x = - 1, + 1$ and $\frac{d^{2}z}{dx^{2}} = \frac{2}{x^{3}}$

$\left( \frac{d^{2}z}{dx^{2}} \right)_{x = 1} = \frac{2}{1} = 2 = + ve$, $\therefore$ $x = 1$ is point of minima.

$x = 1,y = 1$, $\therefore$ minimum value = $x + y = 2$.