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Question Number 207834 by efronzo1 last updated on 28/May/24

$\underset{⏟}{}$
	Answered by Berbere last updated on 28/May/24
	$a^(505) =x;y=b^(505) ⇔ { ((x+y.)),((max{x^4 +y^4 })) :} x,y∈[0,1]^2 ⇒x^4 ≤x;y^4 <y⇒x^4 +y^4 ≤x+y=1 x=1,y=0 x^4 +y^4 =1 max(x^4 +y^4 )=1 min using (((x^4 +y^4 )/2))^(1/4) ≥((x+y)/2)=(1/2) ⇒x^4 +y^4 ≥(1/8)(x=(1/2);y=(1/2))$
	$a^{505} = x; y = b^{505}$ $\Leftrightarrow {\begin{cases} x + y . \\ m a x {x^{4} + y^{4}} \end{cases}$ $x, y \in {[0, 1]}^{2} \Rightarrow x^{4} ⩽ x; y^{4} < y \Rightarrow x^{4} + y^{4} ⩽ x + y = 1$ $x = 1, y = 0 x^{4} + y^{4} = 1$ $m a x (x^{4} + y^{4}) = 1$ $m i n u s i n g \sqrt[4]{\frac{x^{4} + y^{4}}{2}} ⩾ \frac{x + y}{2} = \frac{1}{2}$ $\Rightarrow x^{4} + y^{4} ⩾ \frac{1}{8} (x = \frac{1}{2}; y = \frac{1}{2})$
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