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Solution to puzzle 96: Five real numbers

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| less than or equal to (x.x)½ (y.y)½; thus

|a + b + c + d| less than or equal to (12 + 12 + 12 + 12)½ (a2 + b2 + c2 + d2)½, with equality if, and only if, a = b = c = d; and hence
|7 − e| less than or equal to 2(10 − e2)½

Squaring both sides of this inequality, we obtain

e2 − 14e + 49 less than or equal to 4(10 − e2).

Therefore 5e2 − 14e + 9 = (e − 1)(5e − 9) less than or equal to 0.

One factor must be positive and the other negative; hence 1 less than or equal to e less than or equal to 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.


Remark

This result may also be established using the Root Mean Square - Arithmetic Mean inequality; see Power Mean.


Further reading

  1. A tiny remark about the Cauchy-Schwarz inequality
  2. (Adobe) Portable Document Format "Quickie" Inequalities

Source: Traditional

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