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a-b-c-d-R-such-that-a-b-c-d-2-a-c-b-d-3-a-d-b-c-4-find-a-2-b-2-c-2-d-2-minimum-




Question Number 215005 by Abdullahrussell last updated on 25/Dec/24
 a,b,c,d∈R such that,   (a+b)(c+d)=2   (a+c)(b+d)=3   (a+d)(b+c)=4    find:  (a^2 +b^2 +c^2 +d^2 )_(minimum.)
a,b,c,dRsuchthat,(a+b)(c+d)=2(a+c)(b+d)=3(a+d)(b+c)=4find:(a2+b2+c2+d2)minimum.
Answered by A5T last updated on 25/Dec/24
ac+ad+bc+bd=2  ab+ad+bc+cd=3  ab+ac+bd+cd=4  ⇒Σab=ab+ac+ad+bc+bd+cd=((2+3+4)/2)=4.5  a^2 +b^2 +c^2 +d^2 ≥(((a+b+c+d)^2 )/4)  ⇒4(a^2 +b^2 +c^2 +d^2 )≥a^2 +b^2 +c^2 +d^2 +2Σab  ⇒3(a^2 +b^2 +c^2 +d^2 )≥9  ⇒a^2 +b^2 +c^2 +d^2 ≥3
ac+ad+bc+bd=2ab+ad+bc+cd=3ab+ac+bd+cd=4Σab=ab+ac+ad+bc+bd+cd=2+3+42=4.5a2+b2+c2+d2(a+b+c+d)244(a2+b2+c2+d2)a2+b2+c2+d2+2Σab3(a2+b2+c2+d2)9a2+b2+c2+d23
Commented by Abdullahrussell last updated on 25/Dec/24
 Sir, equality holds with (a,b,c,d)=?
Sir,equalityholdswith(a,b,c,d)=?
Commented by A5T last updated on 25/Dec/24
This is a lower bound, equality may not   necessarily hold.
Thisisalowerbound,equalitymaynotnecessarilyhold.
Commented by Abdullahrussell last updated on 25/Dec/24
 Sir, if a^2 +b^2 +c^2 +d^2 =3    then (a,b,c,d)=?
Sir,ifa2+b2+c2+d2=3then(a,b,c,d)=?
Commented by A5T last updated on 25/Dec/24
The actual minimum value is ≈7.
Theactualminimumvalueis7.
Commented by A5T last updated on 26/Dec/24
Commented by Frix last updated on 26/Dec/24
It exactly is 7 at  a=−(1/2)+((√2)/2)∧b=−(3/2)+((√2)/2)∧c=−(1/2)−((√2)/2)∧d=−(3/2)−((√2)/2)
Itexactlyis7ata=12+22b=32+22c=1222d=3222

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