AM-GM inequality

AM stands for arithmetic mean, the usual average. GM stands for geometric mean, which is √(ab) for two numbers. The rule is AM ≥ GM. The two are equal only when a = b.

Why it is true: a square is never negative. So (√a - √b)² ≥ 0. Open the bracket to get a - 2√(ab) + b ≥ 0. Move the middle term across and divide by 2. That gives (a + b)/2 ≥ √(ab). The inequality also holds for three or more numbers, for example (a + b + c)/3 ≥ ∛(abc).

A small example

A rectangle has perimeter 20 cm. What is the largest area it can have? Call the sides a and b. Then 2(a + b) = 20, so a + b = 10.

• By AM-GM, √(ab) ≤ 10/2 = 5.
• Square both sides: ab ≤ 25.
• Equality needs a = b = 5. So the largest area is 25 cm², from a square.

Try a few pairs to see it: 6 × 4 = 24 and 7 × 3 = 21. Both are smaller than 25. The closer the two sides, the bigger the area. Olympiad questions use this to find a largest or smallest value without any graph.

Where it fits

This goes beyond the Class 10 board syllabus. HBCSE's syllabus for the Mathematical Olympiad names it under algebra, as inequalities. The Mathematics Teachers' Association (India), MTA(I), conducts IOQM, the first stage of the Mathematical Olympiad Programme that HBCSE organises for NBHM.

About the author

Mohit Sardana

Chief Architect

An engineer (B.Tech, Computer Science) with over 20 years of preparing students for entrance exams, including as Dean at FIITJEE and Vice-President at Aakash. He is the chief architect of the TRUpreBoards evaluation engine.

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