Question

The positive number $x$ when added to it's reciprocal gives the minimum value of the sum at $x$ equal to

• A. $1$
• B. $-1$
• C. $-2$
• D. $2$

Answer 1

The correct option is A $1$

Let $z=x+\frac{1}{x}$
$\frac{dz}{dx}=1-\frac{1}{x^2}$
$\frac{d^{2}z}{d{x}^2}=\frac{2}{x^3}$
For minima $\frac{dz}{dx}=0\Rightarrow 1-\frac{1}{x^2}=0$
$\Rightarrow x=\pm 1$ but $f^{\prime \prime }\left(1\right)>0$
Hence at $x=1$ $f\left(x\right)$ will attain its minimum.