Question

If $x_1>0,i=1,2,\dots ,50$ and $x_1+x_2+\dots +x_{50}=50$, then the minimum value of $\frac{1}{x_1}+\frac{1}{x_2}+\dots ..+\frac{1}{x_{50}}$ equals

• A. 50
• B. $(50{)}^2$
• C. $(50{)}^3$
• D. $(50{)}^4$

Answer 1

Answer: (A)

$x_1+x_2+x_3+\dots \dots \dots \dots +x_{50}=50$
$AM\ge HM$
$\frac{x_1+x_2+\dots \dots ..+x_{50}}{50}\ge \frac{1}{\frac{\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\dots .+\frac{1}{x_{50}}}{50}}$
$\Rightarrow \frac{x_1+x_2+\dots \dots \dots +x_{50}}{50}\ge \frac{50}{\frac{1}{x_1}+\frac{1}{x_2}+\dots \dots ..+\frac{1}{x_{50}}}$
$\Rightarrow \frac{1}{x_1}+\frac{1}{x_2}+\dots \dots ..+\frac{1}{x_{50}}\ge 50$
so minimum value of $\frac{1}{x_1}+\frac{1}{x_2}+\dots \dots ..+\frac{1}{x_{50}}=50$