The sum of five real numbers is 7; the sum of their squares is 10. Find the minimum and maximum possible values of any one of the numbers.
Let the numbers be a, b, c, d, and e. Then we have
a + b + c + d + e = 7
a2 + b2 + c2 + d2 + e2 = 10
Without loss of generality, we seek the minimum and maximum possible values of e. Rewrite the above as
a + b + c + d = 7 − e
a2 + b2 + c2 + d2 = 10 − e2
We will use the Cauchy-Schwarz inequality to derive an inequality involving only e.
Letting x = (1, 1, 1, 1), y = (a, b, c, d), by Cauchy-Schwarz we have |x.y| (x.x)½ (y.y)½; thus
|a + b + c + d| (12 + 12 + 12 + 12)½ (a2 + b2 + c2 + d2)½, with equality if, and only if, a = b = c = d; and hence
|7 − e| 2(10 − e2)½
Squaring both sides of this inequality, we obtain
e2 − 14e + 49 4(10 − e2).
Therefore 5e2 − 14e + 9 = (e − 1)(5e − 9) 0.
One factor must be positive and the other negative; hence 1 e 1.8.
We must verify that the minimum and maximum values of e may be attained. From the derivation above, we know that equality occurs if, and only if, a = b = c = d. This guides us to:
The minimum of e = 1 is attained when a = b = c = d = 1.5, e = 1.
The maximum of e = 1.8 is attained when a = b = c = d = 1.3, e = 1.8.
Therefore the minimum and maximum possible values of any one of the numbers are 1 and 1.8, respectively.
This result may also be established using the Root Mean Square - Arithmetic Mean inequality; see Power Mean.