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if-a-b-c-d-e-8-and-a-2-b-2-c-2-d-2-e-2-16-what-is-the-maximal-value-of-a-




Question Number 180641 by mr W last updated on 14/Nov/22
if a+b+c+d+e=8 and  a^2 +b^2 +c^2 +d^2 +e^2 =16, what is the  maximal value of a ?
ifa+b+c+d+e=8anda2+b2+c2+d2+e2=16,whatisthemaximalvalueofa?
Answered by Emrice last updated on 14/Nov/22
a<4
a<4
Commented by Rasheed.Sindhi last updated on 15/Nov/22
a≤4
a4
Commented by mr W last updated on 15/Nov/22
if a=4, then b=c=d=e=0, and  a+b+c+d+e=4≠8, therefore a must  be less than 4.
ifa=4,thenb=c=d=e=0,anda+b+c+d+e=48,thereforeamustbelessthan4.
Commented by Rasheed.Sindhi last updated on 15/Nov/22
Yes sir I understood.
YessirIunderstood.
Answered by a.lgnaoui last updated on 15/Nov/22
4
4
Answered by manxsol last updated on 15/Nov/22
si a=4  f_(1   ) 4+.......=8  Σ(b+c+d+e)=4  f_2 >16  a≠4
sia=4f14+.=8Σ(b+c+d+e)=4f2>16a4
Commented by manxsol last updated on 15/Nov/22
si a=4  f_(1   ) 4+.......=8  Σ(b+c+d+e)=4  f_2 >16  a≠4   determinant ((a,b,c,d,e,),(3,2,1,1,1,8),(9,4,1,1,1,(16)))
sia=4f14+.=8Σ(b+c+d+e)=4f2>16a4abcde3211189411116
Commented by mr W last updated on 15/Nov/22
a≠4, that′s true. but a<4 doesn′t  mean the next value for a is 3, because  a,b,c,d,e musn′t be integer.
a4,thatstrue.buta<4doesntmeanthenextvalueforais3,becausea,b,c,d,emusntbeinteger.
Answered by mr W last updated on 15/Nov/22
to get a as large as possible, the other  4 numbers should be as close to each  other as possible. that means the   other numbers should be equal, say   equal to x. then we have  a+4x=8  a^2 +4x^2 =16  ⇒4a^2 +(8−a)^2 =64  5a^2 −16a=0  ⇒a=((16)/5)=3.2  ⇒x=(6/5)=1.2  that means we get a_(max) =3.2  when b=c=d=e=1.2.
togetaaslargeaspossible,theother4numbersshouldbeasclosetoeachotheraspossible.thatmeanstheothernumbersshouldbeequal,sayequaltox.thenwehavea+4x=8a2+4x2=164a2+(8a)2=645a216a=0a=165=3.2x=65=1.2thatmeanswegetamax=3.2whenb=c=d=e=1.2.
Commented by manxsol last updated on 15/Nov/22
Thanks Mr.W. I am learn.
Commented by mr W last updated on 15/Nov/22
an other method  (little thinking, just applying formula)  a=8−(b+c+d+e)  (8−b−c−d−e)^2 +b^2 +c^2 +d^2 +e^2 −16=0  Φ=8−(b+c+d+e)+λ[(8−b−c−d−e)^2 +b^2 +c^2 +d^2 +e^2 −16]  (∂Φ/∂b)=−1+λ[−2(8−b−c−d−e)+2b]=0  ⇒b=(1/(2λ))+8−(b+c+d+e)  similarly  ⇒c=(1/(2λ))+8−(b+c+d+e)  ⇒d=(1/(2λ))+8−(b+c+d+e)  ⇒e=(1/(2λ))+8−(b+c+d+e)  ⇒b=c=d=e=x, say  (∂Φ/∂λ)=(8−b−c−d−e)^2 +b^2 +c^2 +d^2 +e^2 −16=0  ⇒(8−4x)^2 +4x^2 −16=0  5x^2 −16x+12=0  ⇒x=((8±2)/5)=2 or (6/5)  ⇒a=0 (min) or ((16)/5) (max)
anothermethod(littlethinking,justapplyingformula)a=8(b+c+d+e)(8bcde)2+b2+c2+d2+e216=0Φ=8(b+c+d+e)+λ[(8bcde)2+b2+c2+d2+e216]Φb=1+λ[2(8bcde)+2b]=0b=12λ+8(b+c+d+e)similarlyc=12λ+8(b+c+d+e)d=12λ+8(b+c+d+e)e=12λ+8(b+c+d+e)b=c=d=e=x,sayΦλ=(8bcde)2+b2+c2+d2+e216=0(84x)2+4x216=05x216x+12=0x=8±25=2or65a=0(min)or165(max)
Commented by mr W last updated on 16/Nov/22
Using Lagrange Multiplier method  for finding local maximum or   minimum of a function with more  than one variable under a constraint.
UsingLagrangeMultipliermethodforfindinglocalmaximumorminimumofafunctionwithmorethanonevariableunderaconstraint.
Commented by manolex last updated on 16/Nov/22
what teory formation Φ?
whatteoryformationΦ?
Commented by mr W last updated on 16/Nov/22
https://www.wikihow.com/Use-Lagrange-Multipliers

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