0-lt-x-y-z-lt-1-1-x-1-y-1-z-xyz-Find-min-1-x-xy-1-y-yz-1-z-zx-

Question Number 161900 by HongKing last updated on 23/Dec/21

0 < x; y; z < 1

(1 - x) (1 - y) (1 - z) = xyz

Find :

Ω = \min (\frac{1 - x}{xy} + \frac{1 - y}{yz} + \frac{1 - z}{zx})

Answered by aleks041103 last updated on 24/Dec/21

1 - x - y - z + x y + x z + y z = 2 x y z

\frac{1 - x}{xy} + \frac{1 - y}{yz} + \frac{1 - z}{zx} =

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

= \frac{1 - 2 x y z}{x y z} = \frac{1}{x y z} - 2

W e n e e d t o f i n d m a x o f x y z :

(1 - x) (1 - y) (1 - z) = x y z

\frac{1}{z} - 1 = \frac{x y}{(1 - x) (1 - y)}

\Rightarrow z = \frac{1}{1 + \frac{x y}{(1 - x) (1 - y)}} = \frac{(1 - x) (1 - y)}{(1 - x) (1 - y) + x y} =

= \frac{(1 - x) (1 - y) + x y}{(1 - x) (1 - y) + x y} - \frac{x y}{(1 - x) (1 - y) + x y} =

= 1 + \frac{x y}{1 - x - y + 2 x y} = z (x, y)

\Rightarrow f (x, y, z) = x y z (x, y) =

= x y + \frac{x^{2} y^{2}}{1 - x - y + 2 x y}

f_{x} = (1 + \frac{2 x y}{1 - x - y + 2 x y} - \frac{x^{2} y (2 y - 1)}{{(1 - x - y + 2 x y)}^{2}}) y = 0

f_{y} = (1 + \frac{2 x y}{1 - x - y + 2 x y} + \frac{{x y}^{2} (2 x - 1)}{{(1 - x - y + 2 x y)}^{2}}) x = 0

\Rightarrow {(1 - x - y + 2 x y)}^{2} + 2 x y (1 - x - y + 2 x y) + x^{2} y (2 x - 1) = 0

\Rightarrow {(1 - x - y + 2 x y)}^{2} + 2 x y (1 - x - y + 2 x y) + y^{2} x (2 y - 1) = 0

\Rightarrow x^{2} (2 x - 1) y = y^{2} x (2 y - 1) \Rightarrow x (2 x - 1) = y (2 y - 1) = a

\Rightarrow x, y a r e s o l u t i o n s t o 2 p^{2} - p - a = 0

x = y

{(1 - 2 x + 2 x^{2})}^{2} + 2 x^{2} (1 - 2 x + 2 x^{2}) + x^{3} (2 x - 1) = 0

\dots

Commented by HongKing last updated on 28/Dec/21

thank you dear Sir cool