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Let \(P\) be a point on the graph of the equation \(xyz=1\) in three-dimensional space. Find the minimum possible distance between \(P\) and the origin.

 

I think we're supposed to use AM-GM but I'm not quite sure. Please help :(

 Apr 15, 2020
 #1
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Distance is given by \(d=\sqrt{x^2+y^2+z^2}\)  

 

Symmetry dictates that x = y = z at the nearest point to the origin, so \(d=\sqrt3\times x\)

 

Given that \(xyz=1\)  we have \(x^3=1\) or \(x = 1\)  so minimum distance is \(d=\sqrt3\)

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 Apr 15, 2020
 #2
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thank you!!

caadfo1118  Apr 15, 2020

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