Inequalities: AM–GM
The arithmetic mean is at least the geometric mean.
Key idea
The arithmetic mean is at least the geometric mean.
In depth
For non-negative numbers, the arithmetic mean is at least the geometric mean: (a + b)/2 ≥ √(ab), with equality only when a = b. It follows from (√a − √b)² ≥ 0 and is used to find minima.
Worked example
For x > 0, x + 1/x ≥ 2√(x·1/x) = 2, with equality at x = 1.
Formula
Real-world uses
- Finding the cheapest box that holds a set volume
- Optimisation tasks in design and economics
Common mistake
Using it with negative numbers.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Inequalities: AM–GM?
The arithmetic mean is at least the geometric mean. It is a topic at the expert level in Algebra.
What is the formula for Inequalities: AM–GM?
The key formula is (a+b)/2 ≥ √(ab).
Where is Inequalities: AM–GM used in real life?
Real-world examples: Finding the cheapest box that holds a set volume; Optimisation tasks in design and economics.
What should I learn before Inequalities: AM–GM?
Make sure you are comfortable with Polar Coordinates (Algebra View), the previous topic in Algebra.
What is a common mistake with Inequalities: AM–GM?
Using it with negative numbers.
How can I practise Inequalities: AM–GM?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Inequalities: AM–GM?
Move on to Cauchy–Schwarz Inequality next.
Related expert topics in Algebra
- Linear Programming – Optimise a linear objective over a feasible polygon; maxima occur at vertices.
- Rational Root Theorem – Possible rational roots are ± factors of constant over factors of leading coefficient.
- Fundamental Theorem of Algebra – Every degree-n polynomial has exactly n complex roots counted with multiplicity.
- Binomial Theorem – Expand (a+b)ⁿ with binomial coefficients.
- Partial Fractions – Decompose a rational function into simpler fractions.
- Mathematical Induction – Prove statements for all n by base case and inductive step.