x-y-z-are-positive-real-numbers-if-x-4-y-4-z-4-1-Then-find-the-minimum-value-of-x-3-1-x-8-y-3-1-y-8-z-3-1-z-8-

Question Number 194105 by York12 last updated on 27/Jun/23

x, y, z a r e p o s i t i v e r e a l n u m b e r s i f x^{4} + y^{4} + z^{4} = 1

T h e n f i n d t h e m i n i m u m v a l u e o f

\frac{x^{3}}{1 - x^{8}} + \frac{y^{3}}{1 - y^{8}} + \frac{z^{3}}{1 - z^{8}}

Answered by Subhi last updated on 27/Jun/23

p u t f (x) = \frac{x^{3}}{1 - x^{8}}

f^{^{'}} (x) = \frac{3 x^{2} (1 - x^{8}) + 8 x^{7} (x^{3})}{{(1 - x^{8})}^{2}} a t x = \frac{1}{^{4} \sqrt{3}} (v a l u e t h a t a c h i e v e s t h e e q u a l i t y)

f^{^{'}} (\frac{1}{^{4} \sqrt{3}}) = 2.598

y_{f} - y_{0} = m (x_{f} - x_{0})

f (x) = 2.598 (x - \frac{1}{^{4} \sqrt{3}}) + \frac{{(\frac{1}{^{4} \sqrt{3}})}^{3}}{1 - {(\frac{1}{^{4} \sqrt{3}})}^{8}}

f (x) = 2.598 x - 1.48

a c c o r d i n g t o t a n g e n t t h e o r e m

f (x) + f (y) + f (z) ⩾ 2.589 (x + y + z) - 1.48 \times 3

= 2.589 (\frac{3}{^{4} \sqrt{3}}) - 1.48 \times 3 = 1.48 a p p r o x

Commented by Frix last updated on 28/Jun/23

I think at x = y = z = 3^{- \frac{1}{4}} the minimum is

3^{\frac{9}{4}} 2^{- 3} \approx 1.48058326457

Commented by York12 last updated on 28/Jun/23

t h a n k s b r o

Commented by York12 last updated on 28/Jun/23

s i r I h a v e s e n t y o u a f r i e n d r e q u e s t o n

d i s c o r d

Commented by York12 last updated on 28/Jun/23

Commented by York12 last updated on 28/Jun/23

Commented by York12 last updated on 28/Jun/23

s i r I h a v e s e n t y o u a f r i e n d r e q u e s t o n

d i s c o r d

p l e a s e a c c e p t s i r

Answered by AST last updated on 28/Jun/23

⩾ \frac{1}{3} (x^{3} + y^{3} + z^{3}) (Σ \frac{1}{1 - x^{8}}) ⩾ \frac{1}{3} (Σ x^{3}) (\frac{9}{3 - Σ x^{8}})

{(Σ \frac{x^{8}}{3})}^{\frac{1}{8}} ⩾ {(Σ \frac{x^{4}}{3})}^{\frac{1}{4}} = \frac{1}{\sqrt[4]{3}} \Rightarrow Σ x^{8} ⩾ \frac{1}{3} \Rightarrow - Σ x^{8} ⩽ - \frac{1}{3}

\Rightarrow Σ \frac{x^{3}}{1 - x^{8}} ⩾ \frac{1}{3} (Σ x^{3}) (\frac{27}{8}) = \frac{9}{8} (Σ x^{3})

I t r e m a i n s t o f i n d m i n (Σ x^{3}) \dots U s e L a g r a n g e m u l t i p l i e r s

3 x^{2} = λ 4 x^{3} \Rightarrow 3 = 4 λ x \Rightarrow λ = \frac{3}{4 x} \Rightarrow x = \frac{3}{4 λ}

\Rightarrow \frac{3 \times 81}{256 λ^{4}} = 1 \Rightarrow λ = \sqrt[4]{\frac{3 \times 81}{256}} = \frac{3 \sqrt[4]{3}}{4} \Rightarrow x = y = z = \frac{\frac{3}{1}}{\frac{3 \sqrt[4]{3}}{1}} = \frac{1}{\sqrt[4]{3}}

S o, m i n (x^{3} + y^{3} + z^{3}) h o l d s w h e n x = y = z = \frac{1}{\sqrt[4]{3}}

\Rightarrow m i n (x^{3} + y^{3} + z^{3}) = \frac{3}{\sqrt[4]{27}}

\Rightarrow Σ \frac{x^{3}}{1 - x^{8}} ⩾ \frac{9}{8} \times \frac{3}{\sqrt[4]{27}} = \frac{\sqrt[4]{27^{3}}}{8}

Commented by York12 last updated on 28/Jun/23

t h a n k s s i r

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