By AM-GM, \( y + \frac{1}{y} \geq 2 \), with equality when \( y = 1 \), i.e., \( x = 0 \).
["# Understanding the AM-GM Inequality: Proving ( y + \frac{1}{y} \geq 2 ) (with Equality at ( y = 1 ))", "The Arithmetic Mean–Geometric Mean (AM-GM) Inequality is a foundational result in mathematics, frequently used in algebra, optimization, and inequality solving. One of its elegant applications remains the proof of the inequality:", "[\ny + \frac{1}{y} \geq 2\n]\nwith equality if and only if ( y = 1 ), or equivalently when ( x = 0 ) in related expressions.", "---", "## What is the AM-GM Inequality?", "The AM-GM inequality states that for any set of non-negative real numbers, the arithmetic mean is always greater than or equal to the geometric mean.", "For two positive numbers ( a ) and ( b ), AM-GM gives:", "[\n\frac{a + b}{2} \geq \sqrt{ab}\n]", "Equality holds precisely when ( a = b ).", "---", "## Applying AM-GM to ( y + \frac{1}{y} )", "We apply AM-GM to the two positive terms ( y ) and ( \frac{1}{y} ), both of which are positive for ( y > 0 ):", "[\n\frac{y + \frac{1}{y}}{2} \geq \sqrt{y \cdot \frac{1}{y}} = \sqrt{1} = 1\n]", "Multiplying both sides by 2:", "[\ny + \frac{1}{y} \geq 2\n]", "This inequality holds for all ( y > 0 ).", "---", "## When Does Equality Occur?", "Equality in AM-GM occurs only when the two terms are equal, i.e., ( y = \frac{1}{y} ).", "Solving:", "[\ny = \frac{1}{y} \implies y^2 = 1 \implies y = 1 \quad \ ext{(since ( y > 0 ))}\n]", "At ( y = 1 ), we compute:", "[\ny + \frac{1}{y} = 1 + 1 = 2\n]", "Thus, ( y = 1 ) achieves equality, confirming the sharpness of the inequality.", "---", "## Connection to ( x = 0 )", "In analytical contexts, expressions like ( y = \frac{1}{x} ) arise naturally. When ( x = 0 ), ( y = \frac{1}{0} ) is undefined, but approaching ( x \ o 0^+ ) leads ( y \ o +\infty ), which does not satisfy the inequality.", "However, in the original formulation ( y + \frac{1}{y} \geq 2 ), we restrict ( y > 0 ), and equality occurs only at ( y = 1 ), not at ( y = 0 ) — since ( \frac{1}{y} ) is undefined at ( y = 0 ).", "But if the expression involves ( x = 0 ), consider when ( y = \frac{1}{x} ) and ( x \ o 0 ). While ( \frac{1}{x} \ o \infty ), the expression asymptotically tends toward infinity, far exceeding 2. Hence, equality does not arise at ( x = 0 ); rather, ( y = 1 ) remains the unique point where ( y + \frac{1}{y} = 2 ).", "---", "## Conclusion", "The AM-GM inequality provides a powerful tool to prove:", "[\ny + \frac{1}{y} \geq 2\n]", "with equality exactly when ( y = 1 ). This means ( y <br/>\neq 1 ) yields values strictly greater than 2, illustrating the inequality’s tight bound. The situation ( y = 1 ) corresponds to a critical threshold, while ( x = 0 ) (leading to undefined ( y )) remains outside the domain where the expression is valid—preserving the integrity of the result.", "Understanding this inequality deepens insight into function behavior, optimization, and the elegance of mathematical symmetry.", "---", "Keywords: AM-GM inequality, ( y + \frac{1}{y} \geq 2 ), equality condition, ( y > 0 ), mathematical inequality proof, ( x = 0 ), AM-GM application, inequality analysis."]
