Menu Close

Show-that-a-b-2-2-a-2-b-2-

Tinku Tara 0 Comments
Pin




Question Number 49604 by Tawa1 last updated on 08/Dec/18
Show that: (((a + b)^2 )/2) ≤ a^2 + b^2
Showthat:(a+b)22⩽a2+b2
Answered by afachri last updated on 08/Dec/18
let (a − b)^2 ≥ 0 a^2 − 2ab + b^2 ≥ 0 a^2 + b^2 ≥ 2ab a^2 + 2ab + b^2 ≥ 2ab + 2ab (a + b)^2 ≥ 4(ab) meanwhile : ab = (( (a + b)^2 − (a^2 + b^2 ) )/2) (a + b)^2 ≥ 4((( (a + b)^2 − (a^2 + b^2 ))/2)) (a + b)^(2 ) ≥ 2(a + b)^2 − 2(a^2 + b^2 )] −(a + b)^2 ≥ −2(a^2 + b^2 ) (((a + b)^2 )/2) ≤ (a^2 + b^2 )
let(a−b)2⩾0a2−2ab+b2⩾0a2+b2⩾2aba2+2ab+b2⩾2ab+2ab(a+b)2⩾4(ab)meanwhile:ab=(a+b)2−(a2+b2)2(a+b)2⩾4((a+b)2−(a2+b2)2)(a+b)2⩾2(a+b)2−2(a2+b2)]−(a+b)2⩾−2(a2+b2)(a+b)22⩽(a2+b2)
Commented by Tawa1 last updated on 08/Dec/18
God bless you sir.
Godblessyousir.
Commented by Tawa1 last updated on 08/Dec/18
Sir please help with this too Show that: ((abcd))^(1/4) = (1/4) (a + b + c + d)
SirpleasehelpwiththistooShowthat:abcd4=14(a+b+c+d)
Commented by afachri last updated on 08/Dec/18
Arithmaric Mean = ((a + b + c + d)/4) Geometric Mean = ((abcd^ ))^(1/4) Arithmatic Mean ≥ Geometric Mean (1/4)(a + b + c + d) ≥ ((abcd^ ))^(1/4) so, the equalty can be achieved only and if only : a = b= c = d then (1/4)(a + b + c + d) = ((abcd^ ))^(1/4)
ArithmaricMean=a+b+c+d4GeometricMean=abcd4ArithmaticMean⩾GeometricMean14(a+b+c+d)⩾abcd4so,theequaltycanbeachievedonlyandifonly:a=b=c=dthen14(a+b+c+d)=abcd4
Commented by Tawa1 last updated on 08/Dec/18
God bless you sir
Godblessyousir
Commented by afachri last updated on 08/Dec/18
ur welcome, Sir. Pardon me, are u an Indonesian ?
urwelcome,Sir.Pardonme,areuanIndonesian?
Commented by Tawa1 last updated on 08/Dec/18
No sir
Nosir
Commented by afachri last updated on 08/Dec/18
nevermind Sir. i′m just asking Sir. :)
nevermindSir.i′mjustaskingSir.:)
Commented by Tawa1 last updated on 08/Dec/18
Ok sir
Oksir
Commented by Tawa1 last updated on 08/Dec/18
What if the first question is: (((a+ b)/2))^2 ≤ ((a +b)/2)
Whatifthefirstquestionis:(a+b2)2⩽a+b2
Commented by Tawa1 last updated on 08/Dec/18
How will the prove be
Howwilltheprovebe
Commented by afachri last updated on 08/Dec/18
i had given the 2 solutuions earlier Sir. what kind of proof else you sesrching for Sir ?? i′m sorry i don′t get it.
ihadgiventhe2solutuionsearlierSir.whatkindofproofelseyousesrchingforSir??i′msorryidon′tgetit.
Answered by afachri last updated on 08/Dec/18
according to QM−AM QM ≥ AM (√( (( a^2^ + b^2 )/2) ))≥ (( a + b )/( 2)) then square both sides. (( a^2 + b^2 )/2) ≥ (( (a + b)^2 )/( 4_ )) a^2 + b^2 ≥ (((a + b)^2 )/2)
accordingtoQM−AMQM⩾AMa2+b22⩾a+b2thensquarebothsides.a2+b22⩾(a+b)24a2+b2⩾(a+b)22
Commented by Tawa1 last updated on 08/Dec/18
God bless you sir
Godblessyousir
Commented by Tawa1 last updated on 08/Dec/18
What is QM and AM sir
WhatisQMandAMsir
Commented by afachri last updated on 08/Dec/18
you′re welcome,Sir
you′rewelcome,Sir
Commented by Tawa1 last updated on 08/Dec/18
???
???
Commented by afachri last updated on 08/Dec/18
quadratic mean and arithmaric mean
quadraticmeanandarithmaricmean
Commented by Tawa1 last updated on 08/Dec/18
God bless you sir. I really appreciate your time
Godblessyousir.Ireallyappreciateyourtime
Commented by afachri last updated on 08/Dec/18
it′s been my pleasure Sir
it′sbeenmypleasureSir

Pin

Leave a Reply

Your email address will not be published. Required fields are marked *