if x>0,y>0,z>0 show that ((x+y)/z)+((z+y)/( x))+((z+x)/y)≥6


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Question Number 84915 by M±th+et£s last updated on 17/Mar/20

$i f$ $x > 0, y > 0, z > 0$ $s h o w t h a t$ $\frac{x + y}{z} + \frac{z + y}{x} + \frac{z + x}{y} ⩾ 6$
	Answered by TANMAY PANACEA last updated on 17/Mar/20

	$\frac{x + y}{z} + 1 + \frac{z + y}{x} + 1 + \frac{z + x}{y} + 1 - 3$ $(x + y + z) (\frac{1}{x} + \frac{1}{y} + \frac{1}{z}) - 3$ $u s i n g A M - G M - H M$ $A M ⩾ G M ⩾ H M$ $A M = \frac{x + y + z}{3}$ $H M = \frac{3}{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}}$ $\frac{x + y + z}{3} ⩾ \frac{3}{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}}$ $(x + y + z) (\frac{1}{x} + \frac{1}{y} + \frac{1}{z}) ⩾ 9$ $(x + y + z) (\frac{1}{x} + \frac{1}{y} + \frac{1}{z}) - 3 ⩾ 9 - 3$ $s o$ $(x + y + z) (\frac{1}{x} + \frac{1}{y} + \frac{1}{z}) ⩾ 6$
	Commented byM±th+et£s last updated on 17/Mar/20

	$t h a n k y o u s i r$
	Commented byTANMAY PANACEA last updated on 17/Mar/20

	$m o s t w e l c o m e s i r$
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