a, b, c, d, e are real numbers such that
a + b + c + d + e = 8
a^2 + b^2 + c^2 + d^2 + e^2 = 16.
What is the largest possible value of e?
Let’s suppose $\displaystyle a$, $\displaystyle b$, $\displaystyle c$, $\displaystyle d$ are non-negative. (Obviously $\displaystyle e$ will have to be positive if we want to maximize it.)
Applying AM–GM to the first equation gives
$\displaystyle 8-e=4\left(\frac{a+b+c+e}{4}\right)\ge\sqrt[4]{abcd}$
$\displaystyle \therefore\ e\le8-4\sqrt[4]{abcd}$
and e is maxed when $\displaystyle a=b=c=d$.
This is consistent with the application of AM–GM to the second equation, when we get
$\displaystyle e^2\le16-4\sqrt[4]{a^2b^2c^2d^2}=16-4d^2$ when $\displaystyle a^2=b^2=c^2=d^2$.
(Clearly, for positive e, e is maxed if and only $\displaystyle e^2$ is maxed.)
Solving those two equations in $\displaystyle e$ and $\displaystyle d$ gives $\displaystyle e=\frac{16}{5}$ as the maximum value for $\displaystyle a,b,c,d\ge0$.
If some or all of $\displaystyle a$, $\displaystyle b$, $\displaystyle c$, $\displaystyle d$ are negative, some other method may have to be tried.