Inequalities
AM-GM, Cauchy-Schwarz, triangle inequality, and optimisation problems
Key Concepts
- AM–GM: the arithmetic mean is at least the geometric mean
- Cauchy–Schwarz bounds sums of products
- The triangle inequality and its variants
- Equality conditions are often the key to the problem
Important Formulae
| AM–GM | (a+b)/2 ≥ √(ab), equality when a = b |
| Cauchy–Schwarz | (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²) |
| Power mean / QM–AM | √((a²+b²)/2) ≥ (a+b)/2 |
Quick Tips
- Equality in AM–GM holds only when all the terms are equal.
- Normalising (e.g. setting a+b+c = 1) often simplifies an inequality.
Sample Practice Questions
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If a, b > 0, by AM-GM: a² + b² ≥ ?
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Answer: 2ab
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The inequality a² + b² + c² ≥ ab + bc + ca holds when:
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Answer: Always (for all reals)
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The Arithmetic-Harmonic mean inequality: for positive a, b, AM ≥ HM. HM of a and b = ?
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Answer: 2ab/(a+b)
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For x > 0: eˣ ≥ 1 + x. Equality holds when:
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Answer: x = 0
Practice Questions
Practise PRMO questions on Inequalities. Answers are revealed after each question.
Start Practice →Frequently Asked Questions
What are the key concepts in Inequalities for PRMO?
AM–GM: the arithmetic mean is at least the geometric mean; Cauchy–Schwarz bounds sums of products; The triangle inequality and its variants; Equality conditions are often the key to the problem.
Which formulae are important for Inequalities in PRMO?
AM–GM: (a+b)/2 ≥ √(ab), equality when a = b; Cauchy–Schwarz: (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²); Power mean / QM–AM: √((a²+b²)/2) ≥ (a+b)/2.
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