1-x-1-y-1-z-1-find-the-minimum-value-of-x-2-y-2-z-2-

Question Number 192612 by York12 last updated on 22/May/23

\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1

f i n d t h e m i n i m u m v a l u e o f x^{2} + y^{2} + z^{2}

Commented by Frix last updated on 23/May/23

Yes of course!

Commented by AST last updated on 23/May/23

F o r p o s i t i v e x, y, z; m i n (x^{2} + y^{2} + z^{2}) = 27 a t

x = y = z = 3

Commented by York12 last updated on 23/May/23

e x a c t l y

Answered by Subhi last updated on 23/May/23

A p p l y {t i t u}^{'} s L e m m a e n q u a l i t y

1 = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} ⩾ \frac{9}{x + y + z} ⩾ \frac{9}{\sqrt{3 (x^{2} + y^{2} + z^{2})}}

\sqrt{3 (x^{x} + y^{2} + z^{2})} ⩾ 9

x^{2} + y^{2} + z^{2} ⩾ \frac{81}{3} = 27

t h e n, t h e m i n i v a l u e i s 27

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