Understanding Arithmetic Mean (AM) and Geometric Mean (GM)

For two positive quantities, say \({\rm{x}}_1\) and \({\rm{x}}_2\), the Arithmetic Mean (AM) is defined as:

\({\rm{AM}} = \frac{{{\rm{x}}_1} + {{\rm{x}}_2}}{2}\)

The Geometric Mean (GM) for the same positive quantities is defined as:

\({{\rm{GM}}} = \sqrt {{{\rm{x}}_1}{{\rm{x}}_2}} \)

A fundamental inequality in mathematics states that for non-negative real numbers, AM is always greater than or equal to GM (\({\rm{AM}} \ge {\rm{GM}}\)). Equality holds only when the numbers are equal (\({{\rm{x}}_1} = {{\rm{x}}_2}\)).

Deriving the Condition for AM - GM > 1

The question asks for the condition under which the difference between the arithmetic mean and the geometric mean is greater than 1. Mathematically, this condition is:

\({{\rm{AM}}} - {{\rm{GM}}} > 1\)

Substitute the formulas for AM and GM into this inequality:

\(\frac{{{\rm{x}}_1} + {{\rm{x}}_2}}{2} - \sqrt {{{\rm{x}}_1}{{\rm{x}}_2}} > 1\)

To simplify the inequality, multiply both sides by 2:

\({{\rm{x}}_1} + {{\rm{x}}_2} - 2\sqrt {{{\rm{x}}_1}{{\rm{x}}_2}} > 2\)

We can recognize the left side of this inequality as the expansion of a squared term. Since \({\rm{x}}_1\) and \({\rm{x}}_2\) are positive, we can consider their square roots, \(\sqrt{{{\rm{x}}_1}}\) and \(\sqrt{{{\rm{x}}_2}}\). Recall the algebraic identity \((a-b)^2 = a^2 - 2ab + b^2\). Let \(a = \sqrt{{{\rm{x}}_1}}\) and \(b = \sqrt{{{\rm{x}}_2}}\). Then \(a^2 = {{\rm{x}}_1}\), \(b^2 = {{\rm{x}}_2}\), and \(ab = \sqrt{{{\rm{x}}_1}}\sqrt{{{\rm{x}}_2}} = \sqrt{{{\rm{x}}_1}{{\rm{x}}_2}}\). Thus, the left side is:

\({{\rm{x}}_1} - 2\sqrt {{{\rm{x}}_1}{{\rm{x}}_2}} + {{\rm{x}}_2}} = {\left( {\sqrt {{{\rm{x}}_1}} - \sqrt {{{\rm{x}}_2}} } \right)^2}\)

So the inequality becomes:

\({\left( {\sqrt {{{\rm{x}}_1}} - \sqrt {{{\rm{x}}_2}} } \right)^2} > 2\)

To isolate the term involving the square roots, we can take the square root of both sides. When taking the square root of a squared term, we must consider the absolute value:

\(\sqrt{{{\left( {\sqrt {{{\rm{x}}_1}} - \sqrt {{{\rm{x}}_2}} } \right)}^2}} > \sqrt 2 \)

\(\left| {\sqrt {{{\rm{x}}_1}} - \sqrt {{{\rm{x}}_2}} } \right| > \sqrt 2 \)

Comparing with Options

We have derived the condition \(\left| {\sqrt {{{\rm{x}}_1}} - \sqrt {{{\rm{x}}_2}} } \right| > \sqrt 2 \) for the difference between the arithmetic mean and the geometric mean of positive quantities \({\rm{x}}_1\) and \({\rm{x}}_2\) to be greater than 1.

Let's examine the given options:

• Option 1: \({{\rm{x}}_1} + {{\rm{x}}_2} > 2\sqrt {{{\rm{x}}_1}{{\rm{x}}_2}} \). This simplifies to \(\frac{{{\rm{x}}_1} + {{\rm{x}}_2}}{2} > \sqrt {{{\rm{x}}_1}{{\rm{x}}_2}} \), which is the standard AM > GM inequality for unequal positive numbers. This is a weaker condition than AM - GM > 1.
• Option 2: \(\sqrt {{{\rm{x}}_1}} + \sqrt {{{\rm{x}}_2}} > \sqrt 2 \). This inequality does not directly result from the manipulation of the AM - GM > 1 condition.
• Option 3: \(\left| {\sqrt {{{\rm{x}}_1}} - \sqrt {{{\rm{x}}_2}} } \right| > \sqrt 2 \). This is exactly the condition we derived. Squaring both sides gives \({\left( {\sqrt {{{\rm{x}}_1}} - \sqrt {{{\rm{x}}_2}} } \right)^2} > 2\), which expands to \({{\rm{x}}_1} - 2\sqrt {{{\rm{x}}_1}{{\rm{x}}_2}} + {{\rm{x}}_2}} > 2\), and rearranging gives \(\frac{{{\rm{x}}_1} + {{\rm{x}}_2}}{2} - \sqrt {{{\rm{x}}_1}{{\rm{x}}_2}} > 1\).
• Option 4: \({{\rm{x}}_1} + {{\rm{x}}_2} < 2{\rm{\;}}\left( {\sqrt {{{\rm{x}}_1}{{\rm{x}}_2}} + 1} \right)\). This simplifies to \({{\rm{x}}_1} + {{\rm{x}}_2} < 2\sqrt {{{\rm{x}}_1}{{\rm{x}}_2}} + 2\), which means \({{\rm{x}}_1} + {{\rm{x}}_2} - 2\sqrt {{{\rm{x}}_1}{{\rm{x}}_2}} < 2\), or \({\left( {\sqrt {{{\rm{x}}_1}} - \sqrt {{{\rm{x}}_2}} } \right)^2} < 2\). This is the opposite condition.

Therefore, the condition for the difference between the arithmetic mean and the geometric mean to be greater than 1 is \(\left| {\sqrt {{{\rm{x}}_1}} - \sqrt {{{\rm{x}}_2}} } \right| > \sqrt 2 \).

Conclusion

The condition \(\left| {\sqrt {{{\rm{x}}_1}} - \sqrt {{{\rm{x}}_2}} } \right| > \sqrt 2 \) is equivalent to the difference between the arithmetic mean and the geometric mean of two positive quantities \({\rm{x}}_1\) and \({\rm{x}}_2\) being greater than 1.