if-xyz-1-prove-x-x-1-2-y-y-1-2-z-z-1-2-1-

Question Number 196519 by mr W last updated on 26/Aug/23

i f x y z = 1, p r o v e

{(\frac{x}{x - 1})}^{2} + {(\frac{y}{y - 1})}^{2} + {(\frac{z}{z - 1})}^{2} ⩾ 1 .

Answered by mahdipoor last updated on 27/Aug/23

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

g (x, y, z) = x y z = 1

i f p = (X, Y, Z) i s e x t e r m u m f i n D_{f} = D_{g}

\Rightarrow \vec{▽} f (p) = λ \vec{▽} g (p)

I \Rightarrow 2 (\frac{X}{X - 1}) (\frac{- 1}{{(X - 1)}^{2}}) = λ (Y Z)

I I \Rightarrow 2 (\frac{Y}{Y - 1}) (\frac{- 1}{{(Y - 1)}^{2}}) = λ (X Z)

I I I \Rightarrow 2 (\frac{Z}{Z - 1}) (\frac{- 1}{{(Z - 1)}^{2}}) = λ (X Y)

I V \Rightarrow X Y Z = 1

X, Y = 0, Z = \infty o r e t c

\Rightarrow f (0, 0, \infty) = 1 = m i n i m u m

f ((x, y, z) \in D_{g}) > 1

Commented by AST last updated on 27/Aug/23

f (\frac{1}{4}, - 2, - 2) = 1

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