If-x-y-gt-0-log-x-log-y-2-What-is-the-minimum-value-of-1-x-1-y-

Question Number 108000 by ZiYangLee last updated on 13/Aug/20

If x, y > 0

\log x + \log y = 2,

What is the minimum value of \frac{1}{x} + \frac{1}{y}

Answered by mr W last updated on 13/Aug/20

\ln x y = 2

\Rightarrow x y = e^{2}

A M ⩾ G M

\frac{1}{x} + \frac{1}{y} ⩾ 2 \sqrt{\frac{1}{x} \times \frac{1}{y}} = \frac{2}{\sqrt{x y}} = \frac{2}{e}

i . e . {(\frac{1}{x} + \frac{1}{y})}_{m i n} = \frac{2}{e}

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