Finding Maximum Value of $e$ in $a,b,c,d,e \in R$

[Albert1]

$a,b,c,d,e \in R$
$a+b+c+d+e=8$
$a^2+b^2+c^2+d^2+e^2=16$
$find :\,\, e_{max}$

 

[M R]

$a,b,c,d,e \in R$
$a+b+c+d+e=8$
$a^2+b^2+c^2+d^2+e^2=16$
$find :\,\, e_{max}$

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

 

[Albert1]

$e_{max}=?$
and can you prove it ?

 

[MarkFL]

My solution:

Spoiler

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]

$e_{max}=?$
and can you prove it ?

Yep. See MarkFL's post. :p