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

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The discussion focuses on maximizing the value of \( e \) given the constraints \( a+b+c+d+e=8 \) and \( a^2+b^2+c^2+d^2+e^2=16 \) for real numbers \( a, b, c, d, e \). It is established that when \( a = b = c = d = 1.2 \), the maximum value of \( e \) is 3.2. The participants are encouraged to prove this maximum value through mathematical reasoning and optimization techniques.

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

[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]
 
$e_{max}=?$
and can you prove it ?
 
My solution:

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

$$a=b=c=d$$

And so:

$$4a+e=8\implies a=\frac{8-e}{4}$$

$$4a^2+e^2=16$$

Substitute for $a$:

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

This simplifies to:

$$e(5e-16)=0$$

Hence:

$$e_{\max}=\frac{16}{5}$$
 
Albert said:
$e_{max}=?$
and can you prove it ?

Yep. See MarkFL's post. :p
 

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