Show that for all a,b,c,d ∈ R with a,b,c,d ≥ 0 1) (√(ab))(√(cd)) ≤ (1/4)(a^2 +b^2 +c^2 +d^2 ) 2) (abcd)^(1/4) ≤ (1/4)(a+b+c+d) Help!


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Question Number 193116 by Mastermind last updated on 04/Jun/23

$Show that for all a, b, c, d \in R with$ $a, b, c, d ⩾ 0$ $1) \sqrt{ab} \sqrt{cd} ⩽ \frac{1}{4} (a^{2} + b^{2} + c^{2} + d^{2})$ $2) {(abcd)}^{\frac{1}{4}} ⩽ \frac{1}{4} (a + b + c + d)$ $Help!$
	Answered by Subhi last updated on 04/Jun/23

	$1) 4 (a^{2} + b^{2} + c^{2} + d^{2}) ⩾ {(a + b + c + d)}^{2}$ $a + b + c + d ⩾ 4^{4} \sqrt{a b c d} (A M - G M)$ ${(a + b + c + d)}^{2} ⩾ 16 \sqrt{a b c d}$ $4 (a^{2} + b^{2} + c^{2} + d^{2}) ⩾ 16 \sqrt{a b c d}$ $\frac{1}{4} (a^{2} + b^{2} + c^{2} + d^{2}) ⩾ \sqrt{a b} . \sqrt{c d}$
	Commented by Subhi last updated on 04/Jun/23

	$4 (a^{2} + b^{2} + c^{2} + d^{2}) ⩾ {(a + b + c + d)}^{2}$ ${(a + b + c + d)}^{2} = a^{2} + b^{2} + c^{2} + d^{2} + 2 c d + 2 a b + 2 a c + 2 a d + 2 b c + 2 b d$ $= a^{2} + b^{2} + c^{2} + d^{2} + 2 (a b + a c + a d + b c + b d + c d) (i)$ $a^{2} + b^{2} + c^{2} + d^{2}$ $a^{2} + b^{2} ⩾ 2 \sqrt{a^{2} . b^{2}} = 2 a b (A M - G M)$ $b^{2} + c^{2} ⩾ 2 b c ⇛ c^{2} + d^{2} ⩾ 2 c d$ $a^{2} + d^{2} ⩾ 2 a d ⇛ b^{2} + d^{2} ⩾ 2 b d$ $a^{2} + c^{2} ⩾ 2 a c$ $s u m t h e 5 e q u a t i o n s$ $3 (a^{2} + b^{2} + c^{2} + d^{2}) ⩾ 2 (a b + b c + a c + a d + b d + c d) (i i)$ $f r o m (i), (i i)$ ${(a + b + c + d)}^{2} ⩽ 4 (a^{2} + b^{2} + c^{2} + d^{2})$
	Commented by Mastermind last updated on 04/Jun/23

	$Thank you so much$
	Commented by Mastermind last updated on 04/Jun/23

	$Thank you my boss$
	Commented by Mastermind last updated on 04/Jun/23

	${What}^{'} s AM - GM ?$
	Commented by Subhi last updated on 04/Jun/23

	$A r i t h m e t i c g e o m e t r i c m e a n i n e q u a l i t y$ $i t s p r o o f i s b e l o w ↓$
	Answered by Subhi last updated on 04/Jun/23

	$p r o o f f o r A M - G M$ ${(\sqrt{a} - \sqrt{b})}^{2} ⩾ 0$ $a + b - 2 \sqrt{a b} ⩾ 0$ $\frac{a + b}{2} ⩾ \sqrt{a b}$ $\frac{a_{1} + a_{2} + . . . . . . . . . a_{n}}{n} ⩾^{n} \sqrt{a_{1} . a_{2} . . . . . . a_{n}}$ $\frac{a + b + c + d}{4} ⩾^{4} \sqrt{a b c d}$
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