Minimum and maximum There are five real numbers $a,b,c,d,e$ such that
$$a + b + c + d + e = 7$$$$a^2+b^2+c^2+d^2+e^2 = 10$$
How can we find the maximum and minimum possible values of any one of the numbers ?
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I just wanted to point out that if you don't want to appeal to symmetry there is a Lagrange multiplier argument which isn't too bad: Plugging in the first equation into the second, the last 4 variables satisfy the equation
$$(7 - b - c - d - e)^2 + b^2 + c^2 + d^2 + e^2 = 10$$
Suppose you want to maximize $b$ for example. Then the gradient of $(7 - b - c - d - e)^2 + b^2 + c^2 + d^2 + e^2$ has to be a multiple of the gradient of $b$, which means there is some $\lambda$ such that
$$-2(7 - b - c - d - e) + 2b = \lambda$$
$$-2(7 - b - c - d - e) + 2c = 0$$
$$-2(7 - b - c - d - e) + 2d = 0$$
$$-2(7 - b - c - d - e) + 2e = 0$$
But the last three equations are the same as $-2a + 2c = 0$, $-2a + 2d = 0$, and
$-2a + 2e = 0$, which lead to $a = c = d = e$. Thus the $4$ variables other than $b$ are the same and one can now proceed as in Sivaram's argument.
