Question

The sum of two nonzero numbers is $8$. The minimum value of the sum of their reciprocals is

• A. $\frac{1}{4}$
• B. $\frac{1}{2}$
• C. $\frac{1}{8}$
• D. none of these

Answer 1

Answer: (B)

$\frac{1}{2}$

Let $x$ and $y$ be two numbers
$\Rightarrow x+y=8$
Assume $z$ be be sum of their inverse
$z=1/x+1/y=\frac{x+y}{xy}=\frac{8}{xy}=\frac{8}{x(8-x)}$
For minimum value of $z$
$\frac{dz}{dx}=0=\frac{16(4-x)}{\left(x\right(8-x){)}^2}\Rightarrow x=4$
Hence minimum value of $z$ is $=1/4+1/4=\frac{1}{2}$