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find max(e)

Albert

Well-known member
Jan 25, 2013
1,225
$a,b,c,d,e \in R$
$a+b+c+d+e=8$
$a^2+b^2+c^2+d^2+e^2=16$
$find :\,\, e_{max}$
 

M R

Active member
Jun 22, 2013
51
$a,b,c,d,e \in R$
$a+b+c+d+e=8$
$a^2+b^2+c^2+d^2+e^2=16$
$find :\,\, e_{max}$

When a=b=c=d=2 we get e=0 and when a=b=c=d=1.2 we get e=3.2.
 

Albert

Well-known member
Jan 25, 2013
1,225
$e_{max}=?$
and can you prove it ?
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
My solution:

Because of the cyclic symmetry in the variables, we may let:

\(\displaystyle a=b=c=d\)

And so:

\(\displaystyle 4a+e=8\implies a=\frac{8-e}{4}\)

\(\displaystyle 4a^2+e^2=16\)

Substitute for $a$:

\(\displaystyle 4\left(\frac{8-e}{4} \right)^2+e^2=16\)

This simplifies to:

\(\displaystyle e(5e-16)=0\)

Hence:

\(\displaystyle e_{\max}=\frac{16}{5}\)
 

M R

Active member
Jun 22, 2013
51