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?

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- Jul 18th 2008, 10:52 AMperashreal numbers
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? - Jul 19th 2008, 12:18 AMJaneBennet
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.