Let x, y be two variables and x0, xy=1, then minimum value of x+y is _______________ .

Question

Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .

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2 \text { Given }: xy = 1 \Rightarrow y = 1/x f\left( x \right) = x + 1/x \Rightarrow f'\left( x \right) = 1 - 1/x^2 \text { For a local maxima or a local minima, we must have } f'\left( x \right) = 0 \Rightarrow 1 - 1/x^2 = 0 \Rightarrow x^2 - 1 = 0 \Rightarrow x^2 = 1 \Rightarrow x = $\pm$ 1 \Rightarrow x = 1 ..............\left( \text { Given }: x > 1 \right) rArry=1 \text { Now,} f''\left( x \right) = 2/x^3 \Rightarrow f''\left( 1 \right) = 2 > 0 \text { So, x = 1 is a local minima } . \therefore \text { Minimum value of } f\left( x \right) = f\left( 1 \right) = 1 + 1 = 2

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