Know-x-y-z-R-such-that-2x-4y-7z-2xyz-Find-Min-x-y-z-

Question Number 194257 by tri26112004 last updated on 01/Jul/23

K n o w x, y, z \in R^{+} s u c h t h a t :

2 x + 4 y + 7 z = 2 x y z

F i n d M i n (x + y + z) ¿

Commented by Frix last updated on 01/Jul/23

x = 3 \land y = \frac{5}{2} \land z = 2 \Rightarrow answer is \frac{15}{2}

Commented by tri26112004 last updated on 01/Jul/23

C a n y o u e x p l a i n i t ¿

Answered by mr W last updated on 01/Jul/23

F = x + y + z + λ (2 x + 4 y + 7 z - 2 x y z)

\frac{\partial F}{\partial x} = 1 + λ (2 - 2 y z) = 0 \Rightarrow \frac{1}{2 λ} + 1 = y z

\frac{\partial F}{\partial y} = 1 + λ (4 - 2 x z) = 0 \Rightarrow \frac{1}{2 λ} + 2 = x z

\frac{\partial F}{\partial z} = 1 + λ (7 - 2 x y) = 0 \Rightarrow \frac{1}{2 λ} + \frac{7}{2} = x y

l e t ξ = \frac{1}{2 λ}

y z = ξ + 1

x z = ξ + 2

x y = ξ + \frac{7}{2}

\frac{2}{y z} + \frac{4}{x z} + \frac{7}{x y} = 2

\frac{2}{ξ + 1} + \frac{4}{ξ + 2} + \frac{7}{ξ + \frac{7}{2}} = 2

2 ξ^{3} - 25 ξ - 28 = 0

(ξ - 4) (2 ξ^{2} + 8 ξ + 7) = 0

\Rightarrow ξ = 4 > - 1 ✓

\Rightarrow ξ = \frac{- 4 \pm \sqrt{2}}{2} < - 1 \Rightarrow r e j e c t e d

y z = 5 \dots (i)

x z = 6 \dots (i i)

x y = \frac{15}{2} \dots (i i i)

\Rightarrow x = \sqrt{\frac{6 \times \frac{15}{2}}{5}} = 3

\Rightarrow y = \sqrt{\frac{5 \times \frac{15}{2}}{6}} = \frac{5}{2}

\Rightarrow z = \sqrt{\frac{5 \times 6}{\frac{15}{2}}} = 2

\Rightarrow {(x + y + z)}_{m i n} = 3 + \frac{5}{2} + 2 = \frac{15}{2}

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