three reciprocals

Algebra Level 3

Find the minimum value of x + y + z x+y+z if 1 x + 1 y + 1 z = 1 \dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z} = 1 for non-negative reals x , y , z x,y,z .


The answer is 9.

K Q by K Q , 27, Australia

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1 solution

K Q
Oct 29, 2017

Apply the AM-HM directly to obtain a + b + c ⩾ 9 a+b+c \geqslant 9 , which is achieved when a = b = c = 3 a=b=c=3 , therefore minimum is 9.

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