Given three Real numbers (x,y,z),such that x^2 +y^2 +z^2 =1 maximize x^4 +y^4 −2z^4 −3(√2)xyz


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Question Number 195570 by York12 last updated on 05/Aug/23

$Given three Real numbers (x, y, z), s u c h t h a t$ $x^{2} + y^{2} + z^{2} = 1$ $m a x i m i z e$ $x^{4} + y^{4} - 2 z^{4} - 3 \sqrt{2} x y z$
Commented by Frix last updated on 05/Aug/23

$I think the maximum is \frac{65}{64} when z = \pm \frac{\sqrt{2}}{4}$
Commented by York12 last updated on 05/Aug/23

$p l e a s e p r o v i d e w o r k i n g s$
	Answered by Frix last updated on 05/Aug/23

	$x^{2} + y^{2} = 1 - z^{2} \Rightarrow x^{4} + y^{4} = - 2 x^{2} y^{2} + z^{4} - 2 z^{2} + 1$ $x^{4} + y^{4} - 2 z^{4} - 3 \sqrt{2} x y z =$ $= - 2 x^{2} y^{2} - 3 \sqrt{2} x y z - z^{4} - 2 z^{2} + 1$ $Let t = x y$ $- 2 t^{2} - 2 \sqrt{2} t z - z^{4} - 2 z^{2} + 1$ $\frac{d [- 2 t^{2} - 2 \sqrt{2} t z - z^{4} - 2 z^{2} + 1]}{d t} = 0$ $- 4 t - 3 \sqrt{2} z = 0 \Rightarrow t = - \frac{3 \sqrt{2} z}{4}$ $Inserting$ $- z^{4} + \frac{z^{2}}{4} + 1$ $\frac{d [- z^{4} + \frac{z^{2}}{4} + 1]}{d z} = 0$ $- 4 z^{3} + \frac{z}{2} = 0 \Rightarrow z = \pm \frac{\sqrt{2}}{4} [\lor z = 0_{minimum}^{rejected because}]$ $\Rightarrow maximum is \frac{65}{64} ★$ $We have$ $z = \frac{\sqrt{2}}{4} \land x y = - \frac{3}{8} \lor z = - \frac{\sqrt{2}}{4} \land x y = \frac{3}{8}$ $x^{2} + y^{2} = 1 - z^{2} = \frac{7}{8}$ $The values are easy to calculate$
	Commented by York12 last updated on 05/Aug/23

	$O R Z$
	Commented by York12 last updated on 05/Aug/23

	$w h e r e t o l e a r n i n e q u a l i t i e s$ $a n d a r e y o u u s i n g d i s c o r d$ $I w o u l d l i k e t o i n v i t e y o u t o o u r c o m m u n i t y$
	Commented by Frix last updated on 05/Aug/23
	Sorry I don't use any social media platforms. I studied mathematics & physics decades ago at a university, I don't know where else you could learn these things.
	Commented by York12 last updated on 05/Aug/23

	$o k a y s i r t h a n k s$
	Commented by York12 last updated on 05/Aug/23

	$b u t i f y o u w a n t e d y o u a r e a l w a y s w e l c o m e$ $w e a r e a b i g c o m m u n i t y o f h i g h s c h o l a r s$ $f r o m d i f f e r e n t s p o t s o n t h e e a r t h a n d w e$ $n e e d i n s t r u c t i o n s f r o m a n e f f i c i e n t$ $p e r s o n l i k e y o u s i r$
	Commented by necx122 last updated on 06/Aug/23

	$y o u c o u l d s h a r e t h e l i n k, a f e w o f u s a r e$ $a l s o i n t e r e s t e d .$
	Commented by York12 last updated on 06/Aug/23

	$g r e a t$
	Commented by York12 last updated on 06/Aug/23

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