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 , , , are non-negative. (Obviously will have to be positive if we want to maximize it.)
Applying AM–GM to the first equation gives
and e is maxed when .
This is consistent with the application of AM–GM to the second equation, when we get
when .
(Clearly, for positive e, e is maxed if and only is maxed.)
Solving those two equations in and gives as the maximum value for .
If some or all of , , , are negative, some other method may have to be tried.