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.)](https://www.tinkutara.com/question/Q215005.png)
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](https://www.tinkutara.com/question/Q215007.png)
Commented by Abdullahrussell last updated on 25/Dec/24
![Sir, equality holds with (a,b,c,d)=?](https://www.tinkutara.com/question/Q215009.png)
Commented by A5T last updated on 25/Dec/24
![This is a lower bound, equality may not necessarily hold.](https://www.tinkutara.com/question/Q215013.png)
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)=?](https://www.tinkutara.com/question/Q215014.png)
Commented by A5T last updated on 25/Dec/24
![The actual minimum value is ≈7.](https://www.tinkutara.com/question/Q215015.png)
Commented by A5T last updated on 26/Dec/24
![](https://www.tinkutara.com/question/35850.png)
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)](https://www.tinkutara.com/question/Q215040.png)