AM-GM-HM Inequality
For positive reals, the arithmetic mean is at least the geometric mean, which is at least the harmonic mean, with equality exactly when all the numbers are equal.
What is AM-GM-HM Inequality?
For positive reals, the arithmetic mean is at least the geometric mean, which is at least the harmonic mean, with equality exactly when all the numbers are equal.
Key formula / rule: AM ≥ GM ≥ HM for positive numbers only.
Key points
- Prove the AM-GM inequality for two positive numbers.
- Apply AM-GM to find extrema of algebraic expressions.
Common exam trap
Applying the inequality to terms that can be negative.
Definitions
- Term
AM-GM-HM Inequality
- Meaning
For positive reals, the arithmetic mean is at least the geometric mean, which is at least the harmonic mean, with equality exactly when all the numbers are equal.
- Term
AM-GM-HM Inequality — explanation
- Meaning
This chain converts an optimisation problem into an inequality with a known equality case, which is why it settles many minimum and maximum questions in one line without calculus.
Learning objectives
Prove the AM-GM inequality for two positive numbers.
Apply AM-GM to find extrema of algebraic expressions.
Formulae
- Key point
AM ≥ GM ≥ HM for positive numbers only.
- Key point
Equality holds if and only if all entries are equal.
- Key point
For two numbers, GM2 = AM * HM.
Prerequisites
MAT-U1-SS-T1-S1-C2
Common mistakes
Applying the inequality to terms that can be negative.
Claiming a minimum without checking that the equality case is attainable.
Keywords
AM-GM-HM
Inequality
Practice preview
For any set of n positive real numbers a1, a2, ..., an, which of the following inequalities is always true?…
easy
If x and y are positive real numbers, what is the minimum value of x/y + y/x?…
easy
For positive real numbers a, b, and c, when does the equality AM = GM = HM hold?…
easy